UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES 
 
 GIFT OF 
 
 MRS. JOHN C.SHEDD
 
 Copyright, 1905, 
 By William Francis Magie. 
 
 \:) 

 
 0/C31 
 
 to 
 
 INTRODUCTION. 
 
 vS> 
 
 1. Subject-matter of Physics. The subject of physics deals 
 
 with certain phenomena exhibited by bodies in the material 
 world. Many of these phenomena have been classified and 
 described under general laws, which are expressed in mathe- 
 matical form and can be made the basis of mathematical dis- 
 cussion. Other phenomena, however, which have not yet thus 
 been classified, properly belong to the subject of physics. 
 Broadly speaking, those natural phenomena which either have 
 been classified or which we expect to classify in this way, come 
 under the general subject of physics, and are thus distinguished 
 ,from those which are simply classified by common characteristics 
 which cannot be given mathematical formulation. 
 
 2. Method. The mental process involved in the study of 
 physics begins with the data of experience and ends with the 
 classification or description of a body of phenomena, expressed 
 by a law that may be stated in mathematical form. The first 
 step in the process is the gathering of material by the ob- 
 
 <^J servation of natural phenomena. Our observations are some- 
 __ times made as the phenomena are presented to us in opera- 
 tions occurring in nature and not subject to our control; but 
 ordinarily they are obtained by experimenting upon natural 
 bodies in such a way as to suppress, so far as possible, those 
 phenomena which are not immediately the object of observa- 
 tion, and to make prominent those which we desire to observe. 
 The results of these experiments are considered and an at- 
 tempt is made to discover, in the phenomena presented by 
 them, some general or simple numerical relation from which 
 the existence of a general rule or law may be inferred. 
 
 If such a law is suspected, it is then put in mathematical 
 form, and by purely logical processes, taking it in connection 
 with other known laws of physics, deductions are made from 
 it, which, when interpreted as statements of fact, express re- 
 sults which may be examined by experiment. 
 
 206520
 
 4 INTRODUCTION. 
 
 Experiments are then instituted in order to see whether 
 the results reached by the argument just explained have their 
 counterparts in nature. If it is found that properly, arranged 
 experiments exhibit phenomena agreeing with these predicted 
 results, the hypothetical law from which those results were 
 obtained is confirmed. The amount of verification of this sort 
 which an hypothesis requires, before its confirmation' is fully 
 admitted and it is called a physical law, differs in different 
 cases. No physical law can be confirmed by experiment so 
 completely as to force the mind to accept it in the same way 
 that the mind is forced to accept the propositions of geometry 
 as necessary consequences of the axioms and postulates; but 
 as continued experiment reveals more phenomena which arc 
 necessary results of the hypothesis, and no phenomena which 
 are inconsistent with the hypothesis, our belief in the law be- 
 comes stronger. Many physical laws have been confirmed so 
 frequently that our belief in their truth is just as strong, 
 although based on other grounds, as our belief in the con- 
 clusions of geometry. 
 
 3. Measurement. In order that the physical phenomena 
 which are observed can be connected by a law expressed in 
 mathematical form, they must be measured, Measurement 
 consists in comparison of a quantity with a unit of its own 
 kind. These units may be chosen arbitrarily, or at will, but 
 it is found more convenient to limit the arbitrary choice of 
 units to a few of them which are considered fundamental, and 
 to develop the other units by definition. Those units which 
 are usually chosen as fundamental are the units of length, of 
 time, and of mass. The unit of length may be the ordinary 
 unit, such as the yard or the metre, which, is used in the com- 
 mon transactions of business. In physics, however, by a 
 convention which is almost universally followed, the unit of 
 length is the centimetre. The unit of time may be any unit 
 in which intervals of time are measured: in physics it is 
 commonly the second. The unit of mass will be defined when 
 the concept of mass has been introduced and explained.
 
 MECHANICS. 
 
 4. Space Relations of Forces. The first important scien- 
 tific work in physics was done by the Greek geometer Archi- -n 
 medes (287-212 B. C.). The Greek mathematicians had de- " 
 voted almost all their attention to the study of space rela- 
 tions, and in their hands geometry had attained a high de- 
 velopment. Archimedes, reflecting on the concept of force, 
 undertook an investigation of the space relations of forces. 
 
 When a man exerts his muscles in moving a body or in 
 pushing or pulling a body which he cannot move, he is con- 
 scious of the exertion of force. In many cases, as when he 
 bends a spring or sustains a weight, he interprets his sensa- 
 tion by saying, that the body against which he acts exerts a ^ 
 force in the opposite direction to the one which he is exerting 
 and of equal magnitude. He further believes that the body, 
 when in the same position or condition, is still exerting a 
 force, even though he does not directly perceive the counter- 
 acting force. He thus feels willing to use bodies in certain 
 conditions or positions as the means by which forces may be 
 exerted. 
 
 When a body is sustained in the air by the hand, the force 
 exerted by the hand, as is shown by universal experience, is 
 directed vertically upwards. The force which the body is v .,. . 
 assumed to exert in the opposite direction or vertically down- ; 
 wards, is called its weight. Universal experience has shown 
 that the weight of a body depends on its size and on the 
 material of which it is composed. If different bodies com- 
 posed of the same material and of equal volume are tested by 
 the hand, or in any other way, it is found that their weights 
 are equal. Two of these bodies will have the same weight as 
 a body of the same material and of a volume double that of 
 either one of them. By suitable- combinations of equal 
 weights, any one of which is selected as a unit, a scale of 
 
 forces can be constructed. 
 

 
 C MECHANICS. 
 
 If a body is supported in such a way that it can turn 
 freely in a vertical plane around a horizontal axis, at right 
 angles to that plane, and is then acted on by forces, or more 
 specifically, if it has weights hung upon it at different points, 
 it will generally turn about the axis until it at last comes to 
 rest. It is then said to be in equilibrium. It is natural to 
 suppose, and the supposition is confirmed by common observa- 
 tion, that this condition of equilibrium is determined, not 
 merely by the values of the weights, but by the positions of 
 their points of application. The problem of determining the 
 conditions of equilibrium in this case was the one attacked 
 by Archimedes. To simplify the problem as much as possible, 
 we suppose the body to be a straight rod, itself without 
 weight, so that it can be treated as a line turning on an axis 
 passing through one point in that line. Such a body is called 
 an ideal lever. Weights are suspended at different points on 
 this lever and are shifted about until the lever is in equi- 
 librium. The problem then is to determine the relation which 
 exists between the weights and their distances from the axis, 
 when equilibrium is established. Modern analysis of the 
 argument of Archimedes has shown that it is not possible to 
 solve this problem from abstract principles by an argument 
 which does not somewhere involve the very principle which it 
 is desired to prove. It is therefore sufficient to investigate 
 the conditions of equilibrium by experiment. 
 
 As a result of experiment we conclude that equilibrium 
 will exist when the weights which tend to turn the lever 
 around the axis in one sense are so placed as to balance, or 
 counteract, the weights which tend to turn the lever in the 
 opposite sense. Furthermore, we conclude that the effect of 
 each weight depends on the distance of its point of applica- 
 IY<*I tion from the axis - lf we multiply each weight by the dis- 
 ' tance of its point of application from the axis and collect the 
 
 j ,\ V P roduct9 in two groups, according as the weights tend to turn 
 \ the lever in one sense or the other, we find that, in the case of 
 
 equilibrium, the sum of these products in the one group is 
 equal to the sum in the other group.
 
 MECHANICS. 7 
 
 5. Moments of Force. If the lever is not straight, but 
 bent, so that the points of application of the weights are not 
 in the same straight line, the rule just stated does not hold 
 true. Experiment shows, in this case, that the importance of 
 
 any weight in determining equilibrium does not depend on the ~f . ^ 
 
 product of the weight and of the distance of its point of ap- .4 
 
 plication from the axis, but on the product of the weight and 
 
 of the distance from the axis to the vertical line pajsing 
 
 through the point of application of the weight. This product 
 
 is called the moment of the weight, or, if we take the weight 
 
 simply as a representative force, it may be called thejnoment 
 
 offeree. The condition of equilibrium, in the case of a lever 
 
 oT any~ form, may then be stated by saying, that equilibrium 
 
 w.ill exist when the sum of the moments j?l force, which tend to 
 
 turn the lever in one sense equals the sum of the moments of 
 
 force which tend to turn it in the opposite aense. If we give 
 
 to those moments of force, which tend to turn the lever In 
 
 one sense, the positre_sign, and to those which tend to turn 
 
 it in the opposite sense, the negative sign, the condition of 
 
 equilibrium may be stated by saying, that the algebraic sum 
 
 of the moments of force which act on the lever is equal to 
 
 zero. 
 
 6. Resultant of Parallel Forces. Experiment shows that 
 when a lever is in equilibrium, it may be sustained by a force 
 
 equal to the sum of the weights applied to the lever. The ^.V (^ 
 single force which is equal to this, and so is equal to the sum ' 
 of the weights, is called the resultant of the weights. By a 
 generalization which is justified by experience, any set of 
 parallel forces which are applied to a body so as to produce 
 equilibrium about an axis, which is kept fixed by an opposite 
 force, is said to have a resultant equal to the sum of the 
 parallel forces. 
 
 When the axis is kept fixed by means of rigid connections 
 with bodies which are not perceived to be exerting force, 
 although they may be doing so, the axis is called the fulcrum 
 of the lever. The experiments just described show that we 
 may substitute for the fulcrum a force equal to the resultant
 
 MECHANICS. 
 
 of the' parallel forces acting on the lever, without disturbing 
 the conditions of equilibrium. 
 
 7. Centre of Gravity. If an extended body is hung from a 
 flexible thread attached to a point in it, it will take up a 
 position of equilibrium. In this position it is plain that the 
 moments of force due to the weights of the different parts of 
 the body balance each other around any horizontal axis which 
 passes through the vertical line determined by the thread. If 
 the body is then hung from another point of_it, so that it is 
 again in equilibrium, the moments of force due to the several 
 parts will manifestly conform to the same condition. Ex- 
 periment indicates, and analysis proves, that any_two lines 
 thus determined will cut each other at a definite point. If a 
 horizontal axis is passed through this point in any direction 
 in the body, the body will then be in equilibrium. about that 
 axis. Or,' if a fixed support is placed at that point, the body 
 will be in equilibrium on that support, whatever be its posi- 
 
 jtion. The point thus determined is called the^ccntre of 
 gravity. It is the point at which we may consider the re- 
 Isultant force applied which is equal to the sum of the weights 
 of the different parts of the body. When we speak of the 
 weight of a body as a force, we commonly mean the resultant 
 force thus determined, and its point of application is some- 
 where on the vertical line passing through the centre of 
 gravity. 
 
 8. The Lever. In the common use of the lever, the lever 
 itself is sustained by a fulcrum and two \veights are applied 
 to it, one of which is called the power and the other the 
 weight. The ratio of the weight to the power is called the 
 mechanical advantage of the lever. The distances from the 
 axis passing through the fulcrum to the vertical lines passing 
 through the points of application of the power and of the 
 weight respectively are called the arms of the lever. The 
 general law of equilibrium may hen be stated by saying that 
 the lever is in equilibrium when the power is to the weight in 
 the inverse ratio of their respective lever arms; or when the 
 jHQdjJct of the power^ and the power arm is equal to the 
 product of the wejght^ and the weight arm.
 
 MECHANICS. 
 
 The different sorts of levers of this simple type may best 
 be described by substituting for the fulcrum a force equal to 
 the sum of the power and the weight and applied to the axis 
 vertically upward. These three forces form a system of 
 parallel forces in equilibrium, for any one of which we may 
 substitute a fulcrum without altering the equilibrium of the 
 two remaining forces. If we designate one of the two forces j- *~C ^ 
 originally applied to the lever by X, the other by Y, and the ~sy ^ 
 
 third force or resultant by Z, we may describe the different ^ \ 
 
 sorts of levers as follows: The lever is of the first kind when (/ ' V \ 
 
 cither Xj3r Y is the power, the other being the weighjl The . 
 
 lever is oTthe second Jdnd when X_or_Y is the power and Z is s-\ * 
 the weight. The lever" is of the third kind when__Z is the ^ 
 power and X or Y is the weight. In some of these cases the ,-*. 
 mechanical advantage is greater than 1, so that the weight (3j ^ 
 sustained, or the force exerted which takes the place of the - 
 weight, is greater than the power. In other cases the 
 mechanical advantage is less than 1 or is a proper fraction. 
 
 9. The Principle of Work. The equality of moments on / . / / 
 
 both sides of the axis, which has been found to be the condi- -7 **-* /*~ *^f^" 
 tion of equilibrium, may seem at first sight to be a sort of<*- -*-^ 
 artificial or at least accidental condition. The weights are 
 different and the lever arms are different. The fact that their 
 respective products are equal does not of itself indicate any- 
 thing similar to the equality between cause and effect which 
 is accepted so commonly as a fundamental principle of reason- 
 ing. The condition of equilibrium, however, may be expressed 
 in another form, in which this equality between cause and 
 effect is brought out. 
 
 Suppose the lever in equilibrium under the action of two ^/^ 7 
 forces or weights, and suppose it to turn through a small v 
 
 angle about the fulcrum. To simplify the conditions as much 
 as possible we suppose the lever straight and horizontal. 
 Then if it turns through the small angle o, one of the forces 
 will be lifted through a distance which may be expressed by 
 pa while the other is lowered through a distance expressed by wo. 
 If we multiply these distances by the forces applied at the points
 
 JO MECHANICS. 
 
 which move through these distances, we find from the principle of 
 moments already established, that the two products Pp* and Ww* 
 are equal. 
 
 Now the movement of the point of application of a force 
 through a distance measured along the line of the force is an 
 effect which may be measured by the product of the force and 
 the distance. This productjsjiamed the work done by the 
 force, or if the movement of the point of application is in 
 the direction opposite to that of the force, it is named the 
 work done against the force. The condition of equilibrium in 
 this case may then be stated by saying that equilibrium 
 exists when, for a small displacement of the lever about its 
 axis, the work done by the one force is equal to the work done 
 against the other 
 
 In case the lever is bent, a similar investigation will show 
 that the lever will be in equilibrium when the products are 
 equal which are formed by multiplying the forces by the re- 
 spective vertical distances through which their points of ap- 
 plication rise or sink above or below their original positions. 
 Either of these products is named the work done by the force 
 which_enters in it as a^Eaclor. In so denning the product we 
 use the convention that work done against the force may be 
 considered as negative work done by the force. So, in gen- 
 eral, any lever will be in equijikrjuin when, for a small dis- 
 placement about the fulcrum, the positive work done by 
 some of the forces which act on it is numerically equal to 
 the negative work done J>y the remaining forces, or, when the 
 sum of all the work done by all the forces is equal to zero. 
 
 The condition of equilibrium may otherwise be stated, at 
 least when the forces applied to it are weights, by saying that 
 for any small displacement of the lever from its position of 
 equilibrium the centre_pf gravitv_QL the_jweights_j^iseSy or at 
 least does not sink. That this statement is true in the case 
 of the simple straight lever with two weights may be seen by 
 inspection, and its truth in more complicated cases of the bent 
 lever and of many weights may be demonstrated without 
 difficulty. It is of interest because it was ajiopted hotly by_
 
 MECHANICS. 11 
 
 Galileo and by Huygens as a fundamental principle, the evi- 
 dence for which they did not attempt to give. In their minds 
 its validity was evident. So far as it needed demonstration, 
 they showed that its denial involved the denial of the well 
 known truth that bodies of themselves do not rise or move 
 away from the earth. 
 
 When we are considering the equilibrium of a single lever, 
 the principle of work may seem to be little more than a mere 
 restatement of the principle of moments. It is only when we 
 are dealing with systems of levers that the peculiar advan- 
 tages of the principle of work appear. As an illustration, let 
 us consider two levers so connected that the weight end of 
 the one is applied to the pqwer_end_Qf__the other. This pair -4 t V & 
 of levers will be in equilibrium when a force is applied to the vj 
 power end of the first one and a weight of a certain magnitude 
 to the weight end of the second one. The ratio betweeji_this 
 weight and the_j>ower can be worked out by the aid of the 
 principle of moments applied successively to the two levers, 
 on the supposition that, when the first lever is in equilibrium, 
 the force which its weight end applies to the power end of 
 the second lever is that force which will maintain the first 
 lever in equilibrium by itself. When worked out by the 
 principle of moments we find that equilibrium will exist when 
 the product of the power and thp pnwpr arms ia^egual to the 
 product of the weight and the weight arms. In determining 
 the conditions of equilibrium of this system, or of any more 
 complicated system, by the principle of moments, our 
 knowledge of the several levers which make up the system must 
 be complete. 
 
 If we now consider the work done when the system of 
 levers undergoes a small displacement, we perceive that the 
 forces which are exerted between two levers of the system are 
 equal and opposite at the point of junction, so that no work 
 is done when that point moves. The same being true for 
 every point of junction, the only work that is done is that 
 done by the forces applied to the two__frge ends of the system. 
 Equilibrium will exist when the work done by one__oJL these
 
 1:2 MECHANICS. 
 
 forces is equal and opposite in sign to that done by the other 
 force. The ratio of the weight to the power is therefore the 
 inverse ratio of the distances through which the weight and 
 
 I the power move respectively, and to determine the conditions 
 of equilibrium, no knowledge of the intermediate structure 
 of the system is necessary. If we know_simjgily the vertical 
 distances through which the two ends of the system move, 
 when the system undergoes a small displacement, we are able 
 to determine the conditions of equilibrium. 
 
 10. The Pulley. The pulley is a wheel turning freely on 
 a horizontal axis. Over a groove in the rim of the wheel is 
 passed a flexible cord. Weights may be hung on the two ends 
 of the cord or forces may be applied to them. If the supports 
 of the axle are fixed, the pulley is called a fixed pulley. 
 Whether the two sections of the cord are parallel or not, it 
 is plain from symmetry that the fixed pulley will be in equi- 
 librium when the forces applied to the ends of the cord are 
 equal. This condition of equilibrium may be otherwise 
 readily deduced, l)y considering Uie points on the wheel at 
 which thg cords leave it. as the points of application^ the 
 forces which are applied to the cord, and considering the 
 rjidii_of the wheel drawn from the axis to those points as two 
 level- arms of equaMength. The moments of force on either 
 side of the axis are then plainly equal, when the weights or 
 forces are equal. 
 
 (The mechanical advantage of the fixed pullgy is 1. and its 
 " <e is simply to change"the ^direction o f thejiorcg which is 
 applied to the cord. 
 
 Sometimes the pulley is not fixed on a support, but is hung 
 in a loop of the cord, one end of which is fastened to a fixed 
 . Ct support, while a force is applied to the other end. A weight 
 
 is hung on the frame work in which the axle of the pulley is 
 supported. Such a pulley is called_a_jnovable pulley. The 
 conditions of equilibrium, that is the relative magnitudes of 
 this weight and of the force which must be applied to the 
 cord at its free end to sustain the weight, may be deduced by 
 the method, already employed, of tr^a^g^thejgullev^ as an
 
 MECHANICS. 
 
 vr 
 
 equal armed lever. To simplify our conceptions, we suppose 
 the free end of the cord is to be pulled vertically upward by 
 a force. Then, if the pulley is in equilibrium, we may sub- 
 stitute for the fixed support another force equal to the one 
 applied to the free end of the cord, and also directed vertically 
 upward. The resultant of these two equal forces, and there- 
 fore the weight which they will sustain, is equal to the sum 
 of these forces, or is double either one of them. The^me- 
 chanical advantage of the movable pulley is therefore 2. 
 
 Combinations of fixed and movable pulleys may be made 
 (e. g. the block and tackle) of which the mechanical ad- 
 vantage is greater than 2. In typical cases of this sort several 
 pulleys are mounted together in a single frame work or block, 
 and two branches of the cord are associated with each of the 
 pulleys in the block. The mechanical advantage is conse- . 
 quently twice the number of movable__Duljys, or is equal to/ 
 the number of branches of the cord passing over the movable \ 
 pulleys! 
 
 The principle of work applies also to the case of combina- 
 tions of fixed and movable pulleys. When the system is in 
 equilibrium, if the weight hung on the tree end of the cord is 
 lowered through a certain distance, the block of movable 
 pulleys is lifted through a distance equal^ to that through 
 which the aforesaid weight i^Jowered, divided by the number 
 of branches of the cord associated with the movable pulleys. 
 Thus if the applied weight is lowered through the distance _s, 
 and there are 2n branches of the cord, the distance through 
 which the weighT attached to the fhovable pulleys is lifted is 
 s-r-2n. The work done on the weight P falling through the dis- 
 tance s is Ps. The work done on the weight IF as it is lifted 
 through the distance s-=-2n is Ws-r-gn^ If the system is in 
 equilibrium, these two quantities of work are equal, and .. / . / 
 hence W=2nP, which is the relation already obtained be- _ /* 
 tween tnlTtwo weights. ^^ 
 
 11. Non-parallel Forces. It has been shown that when the 
 forces applied to a lever in equilibrium are parallel, the 
 resultant which passes through the fulcrum is equal to the
 
 ]4 MECHANICS. 
 
 sum of the forces which are applied to the lever. This is not 
 thTjeastTif^the forces applied to the lever are ngt parallel. 
 Equilibrium may still exist when the forces are not parallel, 
 andTnT~condition of equilibrium is still the same, namely, 
 that the moments_ofjforce on either side of the axis are equal. 
 But the resultant of the non-parallel forces, though it passes 
 through the fulcrum, is not equal to the sum of the forces. 
 
 The simplest way in which we can investigate the relation 
 of the resultant of non-parallel forces to those forces, is to 
 examine by experiment the case of two forces applied at one 
 point in different directions. The point to which these forces 
 are applied may be kept in equilibrium by the application at 
 the same point of a force properly determined in magnitude 
 and direction. In the special cases which we can examine we 
 find that this third force, which is plainly equal and opposite 
 to the resultant of the two given forces, may be found by the 
 following rule: From any point as origin draw two lines in ' 
 the directions of the given forces, lay off on those lines lengths 
 proportional to the magnitudes of the forces, and with those 
 lengths as sides, complete a parallelogram and draw its diag- 
 onal from the origin. The direction of this diagonal will be 
 the direction of the resultant and the length of the diagonal 
 will be proportional to the magnitude of the resultant. This 
 
 I rule is called the parallelogram of forces. Special cases of it 
 were cited by Aristotle (384-322 B. C.). It was first clearly 
 stated by Newton (1687 A. D.). The way in which Newton 
 developed it will be given later. 
 
 It may be of interest to examine the way in which this 
 parallelogram law for obtaining the resultant of two forces 
 which are not parallel may be deduced from the principle of 
 moments. 
 
 We consider the case of twojorces, not parallel, applied to 
 ajlever and in equilibrium. It is plain TKatTthe resultant of 
 these forces musfpass through the fulcrum. The problem is 
 to find the direction of this resultant and its magnitude. If 
 the lines of the two forces are prolonged untiOhey cross, it ia 
 also plain that the resultant will be in a line which passes
 
 MECHANICS. 
 
 15 
 
 through the point of intersection. About this point the 
 moments of the two forces and of the resultant are zero. If 
 the resultant is to be in every way equivalent to the two 
 forces, the moment of the resultant about any other point 
 must be equal to the sum of the moments of the two forces 
 about the same point. 
 
 From the point P draw the lines PQ and PS proportional in 
 length to the given forces. From the point S draw the line 
 ST parallel and equal to the line PQ, join PT, choose any 
 point O and draw the lines OP, OQ, OS, OT. The triangles 
 OPQ and OPS are proportional to the moments of the two 
 forces about the point O. Now the triangle OPT is equal to 
 the sum of the other two triangles, for the altitude of the 
 triangle OPT, standing on the base OP, i* equal to the sum 
 of the altitudes of the triangles OPQ and OPS, which stand 
 on the same base. The moment of the force represented by 
 PT is therefore equal to the sum of the i. moments of the two 
 given forces. And it may easily be seen that no other position 
 of the point T but the one determined by the construction 
 already given will make the altitude of *.he triangle OPT 
 equal to the sum of the altitudes of the other two triangles 
 for any arbitrarily chosen position of the point O. The 
 resultant PT thus determined, which is *he same as that 
 determined by the parallelogram law, is therefore the only 
 force which will have a moment about any point equal to the 
 sum of the moments of the two given forces, and it is there- 
 fore the only force which is equivalent in all respects to the 
 two given forces. 
 
 The parallelogram law can be deduced in this way from | 
 the principle of moments. If we accept as fundamental the 
 parallelogram^ law, we may, by a demonstration which is 
 essentially similar, deduce from it the principle of momenta. 
 
 12. Composition and Resolution of Forces. As was shown I 
 in the last section, we may represent a force by a line, the \ 
 direction of the line representing the direction of the force, 
 and the number of units of length of the line representing 
 the magnitude of the force measured in terms of some unit. 
 
 h'
 
 16 
 
 MECHANICS. 
 
 This mode of representation of forces facilitates the study of 
 combinations of forces. The rule for finding the resultant of 
 two forces has already been given. If more than two forces 
 are applied at a point, their resultant can be found by using 
 this rule with two of the forces to find their resultant, using 
 this resultant with another of the forces to obtain J*__ngw^ 
 resultant, and proceeding in this way until all the forceshave 
 been used, and a final resultant is reached. A simpler way 
 of obtaining the same resultant is as follows: From any 
 point as origin, draw a line to represent one of the forces; 
 from the end point of that line draw a line to represent an- 
 other of the forces; from the end point of that line draw a 
 line to represent another of the forces, and proceed in this 
 way until all the forces have been introduced. The result 
 of this operation is a broken line, which may be made the 
 perimeter of a polygon by drawing a line from the origin to 
 the end point of the line representing the last force consid- 
 ered. This line is the resultant of the forces. It may hap- 
 pen that the last point of the line representing the last force 
 considered coincides with the origin. In that case the result- 
 ant of the set of forces is zero, or the forces are in equi- 
 librium. 
 
 It is often important to find two or more forces which 
 have a resultant equal to a given force. This cannot be done 
 unless a sufficient number of conditions are given to which 
 the forces to be found shall conform. In the simplest case, 
 in which two forces are to be found whose resultant is equal 
 to a given force, two conditions determining either the direc- 
 tion or the magnitude of one or both forces must be given. 
 
 I The process of finding these forces is called the resolution 
 of a force. The forces which are found are called the compo- 
 nents of the given force, and the force whose components are 
 found is said to be resolved into them. 
 
 As an illustration of the process of resolving a force into 
 two components, we may take the case of a given force whose 
 components in two given direction are desired. From any 
 point draw a line representing the given force; from the same
 
 MECHANICS. 17 
 
 point draw two lines of indefinite length in the given direc- . 
 
 tions, which must be on either side of the direction of the *s\ V 
 given force; with the line representing the given force as 
 diagonal, construct the parallelogram whose sides are in the I 
 given directions ; the components desired are then given by I 
 the sides of this parallelogram. 
 
 We are often asked to find the component of a force in a 
 given direction or making a given angle with the given force. 
 The problem as thus stated lacks one of the two necessary 
 conditions to enable us to reach a definite result; but when 
 it is stated in this way, it is always assumed that the other \ 
 component, about which nothing is said, is to be taken in a j 
 direction at right angles to the one which is desired. With | 
 this understanding, the required component can be found by . 
 drawing the line representing the given force, and then draw- -/ 
 ing two lines from the same origin, one of which is in the 
 given direction and the other at right angles to it. When 
 the rectangular parallelogram is constructed, on these two 
 directions, about the line representing the given force as 
 diagonal, the side of the parallelogram which is in the given 
 direction is the component desired. If the angle between the 
 direction of the given force F and the direction of the desired 
 component is a, the magnitude of the component is F cos a. 
 
 13. The Inclined Plane. If a perfectly smooth rigid plane 1 
 is inclined to the horizontal at any angle, a weight may be I 
 supported on it by a force directed upward parallel with the 
 surface of the plane. Such a plane is called an inclined plane, 
 and it is a fundamental problem in, mechanics to find the me- 
 chanical advantage afforded by such a plane, that is, to find 
 the ratio of the weight on the plane to the force which will 
 sustain it. 
 
 An interesting solution of this problem, which rests I 
 directly upon fundamental perceptions, and does not involve J 
 the use of any formulated principle of mechanics, was given / 
 by Stevinus. Stevinus' solution is as follows: Let us sup- 
 pose two inclined planes such as the one which has been de- 
 scribed, starting from the same horizontal plane, and meeting
 
 18 
 
 at a summit. Let us suppose, further, a perfectly smooth and 
 flexible, heavy, endless cord hung over the planes. Everyone 
 will admit on inspection of this arrangement that, if the cord 
 is originally at rest, it will remain at rest, that is, its weight 
 will be so distributed that the cord will be in equilibrium. Jf 
 this were not so, the cord would have of itself the power of 
 initiating and maintaining continuous motion around the in- 
 clined planes. Since the part of the cord which hangs below 
 the lowest points of the planes is symmetrical with respect to 
 those points, the condition of equilibrium must be maintained 
 by those parts of the cord which rest on the two planes. Since 
 equilibrium is maintained the forces which these two parts 
 exert in opposite directions are equal. Now the weights of 
 the two parts are proportional to their lengths; and therefore 
 the weights which will be sustained by equal forces on planes 
 of the same height are proportional to the lengths of the 
 planes. 
 
 If one of the two planes is perpendicular, the force ex- 
 erted by the part of the cord which hangs perpendicularly is 
 the weight of that part. It will sustain on the inclined plane 
 a part of the cord equal to the length of the inclined plane. 
 The weight of the perpendicular part of the cord is equal to 
 the force which, applied parallel to the surface of the inclined 
 plane, will sustain on it the inclined part of the cord. The 
 ratio of the weight of the inclined part of the cord to that 
 force is the mechanical advantage of the inclined plane, and 
 it is therefore equal_to^ the ratio of the length of the plane to 
 its height. 
 
 Another solution of the same problem was given by Galileo, 
 which is based on the principle of work. Suppose the in- 
 clined plane constructed and that a weight placed upon it is 
 sustained by a cord, parallel with the surface of the plane, 
 passed over a pulley at the summit of the plane, and attached 
 to another weight of properly chosen magnitude, which hangs 
 vertically. Galileo asserted that when this system of two 
 weights is in equilibrium, a small displacement of the system 
 may be effected without doing any work. Suppose this dis-
 
 MECHANICS. 
 
 to 
 
 hi. / 
 
 his J 
 
 placement effected by raising or lowering the weight which 
 hangs vertically; then the work done by that weight is equal 
 to the weight multiplied by the distance through which it is 
 moved. The weight which stands on the plane is at the same 
 time lowered or raised along the plane by the same distance, 
 and is lowered or raised vertically by a distance which is 
 proportional to that distance in the ratio *>r the height of the 
 plane to its length. Since the work done by the weight on 
 the plane is equal to that weight multiplied by the vertical 
 distance through which it moves, in order chat the work done 
 by the weight on the plane shall be equal to the work done by 
 the other weight, the weight on the plane must be greater 
 than the other one in the ratio of the length of the plane to 
 its height. The same conclusion is therefore reached by this 
 method of argument as was reached by Stevinus by his 
 method. 
 
 We may use this result as a basis from which to demon- 
 strate the parallelogram law. It is, however, more ad- 
 vantageous to assume the parallelogram law, and by its help 
 to demonstrate the same law of the inclined plane. Suppose 
 a weight to rest on the -plane in equilibrium. The weight y 
 itself is a force directed vertically downward. Resolve it into * 
 two components, one perpendicular to the plane, the other 
 parallel with the plane. The component perpendicular to 
 the plane is a force which does not need to be counteracted by 
 any other force applied to the weight. The component par- 
 allel to the plane must be counteracted by an equal and op- 
 posite force, if the weight is to be in equilibrium. From the 
 geometry of the figure, this force may easily be seen to be to 
 the weight in the ratio of the height of the plane to its length. 
 
 14. Other Mechanical Powers. The wheel and axle, the 
 wedge, and the screw are other arrangements by which me- 
 chanical advantage may be obtained, and enter so frequently 
 into the construction of machinery and into ordinary me- 
 chanical operations that they rank with the lever, the pulley, 
 and the inclined plane as mechanical powers. The mechanical 
 advantage afforded by each of them can be worked out by con- 
 
 VY- It
 
 20 MECHANICS. 
 
 sidering them as special cases of the three mechanical powers 
 which have been already studied. 
 
 15. Motion Due to Constant Force. When a weight is not 
 sustained by an opposite force it always falls toward the 
 earth. The rate at which it falls evidently increases, that is, 
 it will fall faster and faster as it nears the earth. The motion 
 of such a body is one of the simplest exhibitions of motion 
 caused- by a force. It attracted the attention of Galileo, and 
 the study of it by him was the first step taken in the direction 
 of enlarging the subject of mechanics to include the study of 
 motions of bodies as well as the study of cases of equilibrium. 
 In order to give an account of the results reached by Galileo, 
 certain preliminary notions connected with motion in general 
 must be considered. 
 
 16. Velocity. When a point moves along a straight line 
 in such a way that it passes over equal distances in any arbi- 
 trarily chosen 'equal times, or, when a point moves in a 
 straight line in such a way that the ratio of the distance 
 passed over to the time occupied by the point in passing over 
 that distance is a constant, the point is said to have a con- 
 stant velocity. When these two conditions, of motion in a 
 straight line and of a constant ratio between the space and 
 the time, are not fulfilled, or when either of them is not ful- 
 filled, the point is said to have a variable velocity. The value 
 of the variable velocity, at any point in the line or path over 
 which the point moves, may be found by supposing the moving 
 point, after it passes through the point on the line at which 
 its velocity is desired, to move on with a constant velocity, 
 equal to that which it has as it passes through the point on 
 the line. This constant velocity will be the velocity at the 
 point on the line. In more mathematical language, this 
 velocity may be found by taking the point on the line as the 
 origin from which distance along the line is to be measured, 
 forming the ratio between the distance traversed by the 
 moving point along the line and the time taken by it to 
 traverse that distance, and obtaining the limit of this ratio as 
 the time which elapses and the space which is traversed ap-
 
 MECHANICS. 21 
 
 proach zero as a limit. The numerical value of this limit is 
 the magnitude of the desired velocity, and the direction of 
 the tangent to the line, at the point on it at which the velocity 
 is desired, is the direction of the desired velocity. If we 
 represent by s the distance which the moving point traverses 
 in the case of constant velocity and by t the elapsed time, the 
 
 velocity v is given by the formula v = . 
 
 The unit of velocity is the velocity of a point moving with 
 constant velocity which traverses 1 centimetre in 1 second. 
 
 17. Acceleration. When a point moves with a variable 
 velocity it is said to have acceleration. If the point moves in 
 a straight line and in such, a way that its velocity changes by 
 equal amounts in any arbitrarily chosen equal times, its ac- 
 celeration is said to be constant. When these two conditions 
 are not fulfilled, or when either of them is not fulfilled, the 
 acceleration of the point is variable. In the case of constant 
 acceleration, its numerical value is found by dividing the 
 change in velocity v v', which occurs in any arbitrarily 
 chosen time t, by that time, the quotient obtained being 
 the desired acceleration. If we represent acceleration by a, 
 
 the formula by which it is defined and measured is = v ~ v . 
 
 According as the change in velocity is an increase or a de- 
 crease, the acceleration is positive or negative. 
 
 18. Effect of a Constant Force. Heavy bodies evidently 
 fall in straight lines and with increasing velocity. Their 
 motion is therefore accelerated. The problem taken up by 
 Galileo was the determination of the law according to which 
 the velocity of falling bodies changes. It was out of the ques- 
 tion, with his appliances, to investigate this question by a 
 direct measurement of the velocities of the falling bodies at 
 different instants. It was therefore necessary to attack it 
 indirectly. 
 
 After making an erroneous hypothesis respecting the way 
 in which the velocity changes and rejecting it for what seemed 
 to him good reasons, Galileo assumed that the velocities are
 
 22 
 
 MECHANICS. 
 
 proportional to the times during which the body has been 
 falling. He then reasoned as follows: 
 
 Let us suppose that a falling body has a constant accele 
 ation. When it is first released it starts from rest and i 
 initial velocity (v' in the formula) is zero. After the lapse 
 of * seconds its velocity is v. In the t seconds it will 
 traverse a distance s which is equal to the distance which it 
 would traverse in t seconds if it were moving with a con- 
 stant velocity equal to the average of the velocities with which 
 it actually moves. On the hypothesis that its acceleration is 
 constant and therefore that its velocity has been increasing 
 at a uniform rate, this average velocity is |, and the distance 
 
 vt 
 which it will traverse in the t seconds is therefore s = _ 
 
 Since v = at, we have also s lot 2 . 
 
 That this conclusion is correct may be seen from the fol- 
 lowing considerations: Let us suppose that the change in 
 velocity takes place not uniformly, but by small equal incre- 
 ments occurring at the ends of small equal intervals of time. 
 Construct a diagram by marking off along a horizontal line 
 equal distances representing equal increments of time, and by 
 drawing, from the points thus determined, vertical lines repre- 
 senting the velocities which the point will assume at the cor- 
 responding instants. On the lines thus determined construct 
 parallelograms. The area of each of these parallelograms 
 will be numerically equal to the product of a time interval 
 and of the velocity which the point possesses during that 
 interval, and will therefore be the distance which is traversed 
 by the moving point during that interval. The sum of all these 
 areas is therefore equal to the entire distance traversed by the 
 moving point in the time t under the given conditions. As the 
 time intervals are taken smaller and smaller and the changes 
 in velocity become correspondingly smaller and smaller, the 
 motion of the point becomes more and more nearly that of a 
 point moving with constant acceleration and the area becomes 
 more and more nearly equal to the area of the triangle whose
 
 MKCHANICS 23 
 
 base is the line which is numerically equal to t and whose 
 altitude is the velocity of the point at the end of the time t. 
 In the limit, as the time intervals vanish, the area which repre- 
 sents numerically the distance traversed by the moving point 
 becomes the area of the triangle, and since this area is equal 
 to half the base multiplied by the altitude of the triangle, we 
 reach the formula for the distance traversed which has already 
 been given. 
 
 This relation between the distance traversed by the fall- 
 ing body and the elapsed time may be tested by experiment, 
 and according as the results of experiment exhibit this rela- 
 tion or not, the hypothesis from which it was deduced is 
 confirmed or disproved. 
 
 Galileo's experiment consisted in allowing a smooth brass 
 ball to roll down an inclined plane, and in determining the 
 times taken by the ball to cover different distances. He 
 measured the time by means of a large vessel, filled with 
 water, in the bottom of which was a small opening. This 
 opening was stopped with the finger until the ball was re- 
 leased. At the moment the ball was released, the finger was 
 removed, and the water allowed to now out into a small cup 
 until the ball passed a marked point on the plane, when the 
 opening was closed again. The water that flowed into the 
 cup was weighed, and the time which had elapsed Avas taken 
 proportional to its weight. By making a large number of 
 experiments of this kind, to eliminate experimental errors in 
 the final averages taken, Galileo was able to show that the 
 distances passed over by the ball after starting from rest 
 were proportional to the squares of the times during which 
 it was moving. 
 
 In this way it was shown that the acceleration given to 
 falling bodies by their own weights acting as forces is con- 
 stant. 
 
 Galileo also showed that the acceleration of all falling 
 bodies which are heavy enough to move through the air, 
 without their motion being seriously affected by the resist- 
 ance of the air, is the same. He did this by allowing differ-
 
 24 MECHANICS. 
 
 ent bodies to fall from the top of the Leaning Tower of Pisa, 
 releasing them at the same instant and observing that they 
 reached the ground together. This experimental result he 
 illustrated by the following argument: If two exactly sim- 
 ilar bodies, like two coins, are released at the same instant, 
 they will fall to the ground in the same time, and it does not 
 seem reasonable to suppose that their motions will be in any 
 way different if they are joined together so as to form one 
 body. 
 
 The laws of falling bodies are well illustrated by the use 
 of an apparatus called the Attwood's machine. This con- 
 sists essentially of an easily running pulley, over which "is 
 passed a cord sustaining two equal weights. This system, 
 which is in equilibrium, is set in motion by a small over- 
 weight hung on one end of the cord. By suitable arrange- 
 ments the movements of this system may be examined. We 
 may show that the velocity of the system, if it starts from 
 rest, increases proportionally to the time, and that the dis- 
 tance traversejUby^the system is proportional to the square 
 of the time. 
 
 From the two formulae v=at and s=$at 2 , by eliminating 
 t, we obtain the relation v 2 =2as. This relation is not only 
 useful in solving* problems relating to falling bodies, but will 
 enable us in the future to draw a most important conclusion. 
 
 19. Motion on an Inclined Plane. When a body moves 
 down an inclined plane, the force which moves it is the com- 
 ponent of its weight which is parallel to the plane. We as- 
 sume, as Galileo did, that its acceleration on the plane will be 
 to its acceleration if it is falling freely as this component is 
 to its whole weight. If we represent by p the height of the 
 plane, by s the length of the plane, by a its acceleration on the 
 plane and by g its acceleration when falling freely, we have 
 the relation as=gp>.- Starting with this relation we may prove 
 the following theorem: The times of descent down inclined 
 planes of the same height are proportional to their lengths. 
 For we have s=lat* and hence s z =^gpt i , from which, since g
 
 MECHANICS. 25 
 
 is a constant and p is the same for all planes of the same 
 height, we have t proportional to s. 
 
 * r \Ve may also prove the following theorem: The times of 
 descent are equal down inclined planes whose lengths and 
 inclinations are determined by the chords of a vertical circle 
 drawn from its lowest point. That is, the time of descent 
 down any chord of a vertical circle drawn from a point on 
 the circle to its lowest point is the same as the time of free 
 fall through the diameter of the circle. For, from the formula 
 s'^lgpt 2 , which was obtained in the last paragraph, the time ^/-L.^ / 
 
 t is the same for all inclined planes for which 1 is the same, 
 
 P 
 
 From similar triangles in the circle we have ^ = , or 
 
 s d 
 
 d , and hence d = IgP. The time t is thus the time of 
 
 P 
 
 free fall through the diameter of the circle, and is the same 
 for all the planes which conform to the given conditions. 
 
 - 'We may also prove the following theorem: The velocities 
 acquired in the descent of a body down different inclined 
 planes are equal if the heights of the planes are equal. For, 
 from the formula v*=2as and the relation as=yp, we obtain 
 the formula v 2 =2gp, which shows that the acquired velocity 
 depends only on the height of the plane and not on its length, 
 and is therefore the same for all planes of equal height. This 
 last relation holds even if the path of the moving body is 
 curved, as for example, if it is the arc of a circle; for, if not, 
 the velocity acquired will either be greater or less than the 
 velocity with which the body must start up an inclined plane 
 in order to bring it to the level from which it started to fall. 
 If the velocity thus acquired is greater, the moving body under 
 the action of gravity alone will rise to a higher level than that 
 from which it starts, and this conclusion i* so contrary to our 
 experience that we consider the hypothesis upon which it is 
 based to be erroneous. If the velocity acquired is less, it is 
 only necessary to reverse all the motions in order to reach the 
 same erroneous conclusion from this supposition. We con-
 
 2(1 MECHANICS. 
 
 elude therefore, that when a body falls through any path, the 
 velocity which it acquires in falling depends only on the per- 
 pendicular distance through which it falls and not on the 
 length or shape of the path. 
 
 ^20. Composition of Velocities and Accelerations. Galileo 
 perceived or assumed that when a body is in motion and is 
 then acted on by a force, the motion which it will subsequently 
 have will be obtained by superposing on its original motion 
 the motion given to it by the force. That is, the motion of the 
 body is conceived of as consisting of two motions, which exist 
 in the body independently of each other. 
 
 Let us suppose that a point is at one instant in a certain 
 position and at a later instant in another position. The 
 straight line which joins the first position to the second is 
 called the displacement of the point. This displacement 
 manifestly has a determinate direction as well as a magni- 
 tude, and it has no other characteristics. It is fully deter- 
 mined when the direction and magnitude are given. A 
 quantity which conforms to these conditions is called a 
 vector. 
 
 Tf a point undergoes two successive displacements, the 
 final displacement which is the result of the two, is- obtained 
 by drawing a line from the position first occupied by the 
 displaced point to the position last occupied by it. This 
 line is called the resultant displacement. It is manifestly 
 the diagonal of the parallelogram of which the two displace- 
 ments of the point are sides. What is true of displacements 
 is also true of any similar vectors. The resultant of two 
 or more vectors may be found,, or a vector may be resolved 
 into components, by the rules already given for compounding 
 and resolving forces. 
 
 Velocities, measured by displacements occurring in equal 
 intervals of time,, are vectors, and accelerations, measured 
 by changes in velocity occurring in equal intervals of time, 
 are also vectors of another sort. Velocities and accelera- 
 tions may therefore be compounded and resolved by the
 
 MECHANICS. 27 
 
 general rules which hold for the composition and resolution 
 of vectors. 
 
 21. Projectiles. When a body is thrown off in any direction 
 as a projectile, its motion may be considered as made up of two 
 independent motions, the motion originally given it, which is 
 in the original direction of projection, and the motion im- 
 parted to it by the action of its own weight upon it, which is 
 directed vertically downward. 
 
 We shall first consider the simple case of horizontal pro- 
 jection. Suppose a body to be projected horizontally, from a 
 point taken as origin, along the ar-axis with the velocity u. 
 At the end of the time t its displacement in the direction of 
 the a?-axis will be x=ut. From the instant of projection it 
 begins to fall toward the earth, and if we consider the t/-axis 
 to be directed vertically downward, the distance which it will 
 fall in the time t in the direction of the t/-axis is given by 
 y=.\gF. On the supposition which we have made, that the two 
 motions can exist together in the body without mutual inter- 
 ference, we may obtain simultaneous values of x and y by 
 eliminating t between these equations. The equation obtained 
 
 is y = jj-^ x 2 - This is the equation of a. parabola with its 
 
 vertex at the origin and represents the path of the projectile. 
 
 Since any projectile thrown obliquely upwards will reach 
 a highest point of its path, and will at that point be moving 
 'horizontally, with a velocity equal to the horizontal com- 
 ponent of the original velocity of projection, the path which 
 it will subsequently traverse will be a parabola similar to 
 the one already determined. Because of the symmetry of 
 the conditions, the path of the projectile ^before it reaches 
 its highest point will be the other branch of the same para- 
 bola. 
 
 To investigate the movement of a projectile in this more 
 general case, we suppose it to be projected from the origin <- v 
 obliquely upwards with a velocity V, whose horizontal com- * 
 
 ponent along the horizontal a?-axis is u and whose vertical 
 component along the i/-axis, directed upward, is v. Then in
 
 28 MKCHANICS. 
 
 the time t the distance traversed along the a?-axis is x=ut. 
 The distance which would be traversed along the i/-axis in the 
 same time, if it were not for the change of velocity in that 
 direction due to the weight of the body, is vt. Owing to the 
 action of the weight of the body, the distance actually 
 traversed in the direction of the y-axis is less than this, by 
 the distance through which the body will fall in the time t. 
 It is therefore given by y=vt\gt 1 . Eliminating t between 
 these equations we obtain the equation y = - x ^ * 2 as the 
 
 equation of the path of the body. It is the equation of a 
 parabola which passes through the origin. 
 
 The time of flight is the time taken by the projectile, after 
 leaving the origin, to fall to the same level again. It may be 
 determined by setting y=Q and determining the value of t 
 
 6 2n 
 
 obtained on this condition. We obtain t = 
 
 9 
 
 The range of the projectile is the distance between the 
 origin and the point at which the projectile again meets the 
 #-axis. It is therefore obtained by setting y=Q, and determin- 
 ing the value of x obtained on this condition. The range thus 
 
 obtained is x f = . and is the same, whatever the values of 
 
 9 
 
 u and v may be, when they are such that their product is con- 
 stant. For a given value of V, the initial velocity, there are 
 two directions of projection for which the product uv has the 
 same value. These two directions are equally inclined to the 
 line which bisects the right angle between the two axes. 
 
 For a given value of V the inclination of projection for 
 which the greatest range is obtained is the inclination of this 
 bisector. For, if the angle of inclination of V with the tf-axis 
 is o, we have == V cos a, v = V sin a, and 2uv = 2 V 2 sin a cos a 
 = V*sin 2a ; and this is a maximum for a = 45. 
 
 22. The Pendulum. An arrangement which, for practical 
 reasons, attracted special attention in the early days of the 
 study of mechanics is the pendulum. In the form in which it 
 was first studied, the pendulum consists of a small heavy body,
 
 MECHANICS. 29 
 
 which may be considered a heavy point, swinging at the end 
 of a long light thread firmly fastened at the top. Galileo 
 concluded from observations that the time required for a 
 pendulum to execute one swing is the same, whatever the 
 extent of the swing may be. This conclusion is not strictly 
 correct. Such observations as Galileo could make were not 
 sufficiently accurate to detect its falsity, but the fact is that 
 the time of swing of a pendulum is greater when the extent of 
 the swing is greater. So long as the extent of the swing is 
 kept within certain limits, so that the largest angle made by 
 the suspending thread with the verticul i.s not greater than 10, 
 the times of swing are practically independent of their extent. 
 
 Galileo recognized the value of the pendulum as a means 
 of measuring small intervals of time. Doubtless it was this 
 use to which the pendulum could be put which directed 
 special attention to it. 
 
 Huygens applied the pendulum to the regulation of 
 clocks, and constructed clocks which are in all essential par- 
 ticulars the same as those of the present day. He was led 
 by his use of the pendulum to investigate its properties. 
 
 Huygens found that the property of isochronous oscilla- 
 tion which is possessed approximately by the ordinary pen- 
 dulum swinging in a circular arc, is possessed exactly by a 
 pendulum which is so adjusted as to swing in a cycloidal arc. 
 For this reason the cycloid is called the tautochrone. A 
 body let fall from any point on the arc of a cycloid will 
 reach the lowest point of the arc in the same time. 
 
 The problem presented by the circular or simple pendu- 
 lum is to find the way in which the time of swing or of 
 oscillation depends upon the length of the pendulum. 
 
 It will be shown hereafter that the time of oscillation of a 
 
 pendulum of length r is given by the formula t = IT P*. Since, 
 
 +Jff 
 for any one place on the earth, the value of g is constant, the 
 
 times of oscillation of pendulums of different lengths are 
 proportional to the square roots of the lengths. At different 
 points on the earth's surface, where the values of g are differ-
 
 3,) MECHANICS. 
 
 ent the times of oscillation of the same pendulum are inversely 
 as the square roots of the value of g. Such a difference in the 
 time of oscillation of a pendulum, due to the different values 
 of g at different places, was first observed by Richer when lie 
 transported a carefully rated clock from Paris to Cayenne 
 (1071-1673). 
 
 23. The Physical Pendulum. Real pendulums, especially 
 those used in the regulation of clocks, are not made to con- 
 form to the definition of a pendulum which has been given. 
 A real pendulum is a body of considerable mass, which is often 
 irregularly shaped. If we consider each part of this pendulum 
 as being itself the bob of a- simple pendulum, it is plain that 
 the times of oscillation of these different parts, if they were 
 free from one another, would be very different. As they are, 
 however, bound together into one rigid body, some of them are 
 forced to move faster and others are forced to move slower 
 than they would if left to themselves. Manifestly there is 
 some one of the parts which will swing in the same time as it 
 would if left to itself, or as if it were the bob of a simple 
 pendulum, whose length is equal to its distance from the axis 
 about which the pendulum swings. The problem of finding 
 this length, which is called the length of the equivalent simple 
 pendulum, was solved by Huygens. He employed, as the basis 
 of his solution, the principle that the centre of gravity of a 
 system of bodies which are left to themselves can never rise to 
 a higher level than that from which it started. This principle 
 was questioned by Huygens' contemporaries, and its applica- 
 tion to the solution of the pendulum problem anticipates cer- 
 tain conceptions which may better be brought out in another 
 connection. We shall therefore postpone the discussion of 
 Huygens' result until later. 
 
 24. Newton's Laws of Motion. We have now reached cer- 
 tain fundamental conceptions respecting the motion of a body 
 when subjected to the action of a force. These conceptions, 
 even when not formally expressed, have been implied in all our 
 previous discussions. One of these conceptions is, that motion, 
 as well as vest, is a natural state of a body, and that a body
 
 MKCHAXICS. 81 
 
 will persist in its motion without change except in so far as it 
 is acted on by a force. This conception may be called the 
 principle of inertia. The way in which it was reached by Gali- 
 leo is of special interest, although his process of thought 
 cannot be called a proof of the principle. Galileo conceived a 
 body to move down an inclined plane and at the bottom of 
 the plane, by a change in direction which did not involve a 
 change in the body's velocity, to begin an ascent on a second 
 inclined plane. In accordance with the propositions we have 
 already examined, the body will move along the second in- 
 clined plane over a length which will bring it to the same 
 level as that from which it started to fall. The time of ascent 
 through this length will be proportional to the length. Xow, 
 as the inclination of the second plane is made less and less, 
 the time of ascent w r ill become longer and longer and the 
 retardation of the body, or its change of velocity in a unit 
 time, will be less and less. In the limiting case, in which the 
 inclination of the plane is zero, the time of ascent becomes 
 infinite and there is no retardation, in this special case, there- 
 fore, in which the moving body is not acted on by any force, 
 its motion will be uniform and indefinitely continued. 
 
 Another of these fundamental conceptions is that the result 
 of the application of a constant force to a body is the produc- 
 tion of a constant acceleration. The constant force considered 
 by Galileo was the weight of the body or a component of its 
 weight. Galileo assumed that the accelerations produced in 
 the same body by different components of its weight, that is, 
 the accelerations of the body when moving down planes of 
 different inclinations, are proportional to those components. 
 The fundamental conceptions of force and of acceleration as 
 produced by the action of a force were thus brought into rela- 
 tion. This relation, however, was not stated, or even assumed, 
 as a general one involving the action of all kinds of forces, 
 because the forces considered by Galileo were all of one kind. 
 
 In his Philosophiae Naturalis Principia Mathematica, pub- 
 lished in 1687, Newton collected the scattered and imperfectly 
 formulated principles of his predecessors, and adding to them
 
 32 MECHANICS. 
 
 an additional principle, stated them in three laws of motion, 
 which furnish a complete basis for the science of mechanics. 
 These laws are as follows: 
 
 Law I. Every body continues in its state of rest or of uni- 
 form motion in a straight line, except in so far as it is com- 
 pelled by an external force to change that state. 
 
 Law 11. Change of motion is proportional to the external 
 force applied and takes place in the direction of the applied 
 force. 
 
 Law III. To every action there is an equal and contrary 
 reaction; or, the mutual actions of two bodies are always 
 equal and oppositely directed. 
 
 The first law simply states the principle of inertia as it was 
 understood by Galileo. It is in a sense contained in the second 
 law, for it is an evident implication of the second law that, 
 if no external force is applied to a body, the motion of the 
 body will not change. 
 
 The second law may be considered as describing the way 
 in which the motion of a given body depends upon the action 
 of forces of any kind upon it. There is no limitation, in the 
 statement of the law, to forces which arise in any way from 
 the weight of the body. The forces which act and which cause 
 the change of motion may be due to any cause whatever. By 
 the phrase, change of motion, is meant the change of velocity 
 of the body upon which the force acts, or perhaps better, the 
 acceleration of the body. 
 
 From this second law we may deduce the parallelogram of 
 forces; for, when two forces act on the body at once they 
 will give to the body accelerations in the directions of the 
 forces and proportional to them. These accelerations are 
 vectors and may be added by the parallelogram law, and since 
 the forces which produce them are proportional to them, and 
 in their directions, the forces are also vectors and may be 
 added by the same law. This so-called proof of the parallel- 
 ogram of forces is not logically more complete or convincing 
 than the direct experimental proof which has already been
 
 MECHANICS. 33 
 
 | 
 
 given, for it rests ultimately on the experimental evidence for 
 the truth of the second law. 
 
 That part of the second law which states that the forces 
 which act on a body are proportional to the accelerations 
 which they produce, may be illustrated by the Attwood's 
 machine. If we change the proportionality between force and 
 acceleration into an equation by introducing a constant or 
 factor of proportion, we have the relation F=ma. The ex- 
 periments with the Attwood machine show that the factor of 
 proportion m is proportional to the weight of the system. 
 
 The fact that bodies of different weights fall toward the 
 ground with the same acceleration confirms this conclusion. 
 For if, in the equation F=mg, which applies to each of these 
 different bodies, the acceleration g has a common value and the 
 force F is equal to the weight of the body in each case, the 
 factor of proportion m must be proportional to the weight of 
 the body in each case. When we confine our attention to bodies 
 of the same material but of different sizes, we perceive that 
 the quantity of matter in each of these bodies is proportional 
 to the size or volume of the body, and in special cases we may 
 increase our confidence in this conclusion by observing that 
 special effects produced by the bodies are also proportional to 
 their volumes. In particular, the weights of these bodies are 
 proportional to their volumes. In any such case, then, the 
 factors of proportion for the different bodies are proportional 
 to the quantities of matter contained in them. The conclusion, 
 however, would not be warranted from such observations that 
 the factors of proportion, in the case of bodies of different 
 materials, are a measure of the quantities of matter in the 
 bodies. 
 
 To draw this conclusion we must consider the third law, 
 by means of which a valid measure may be obtained of the 
 quantity of matter contained in a body, so far as it relates to 
 the action of force upon that body. 
 
 Before proceeding to the discussion of this point, another 
 feature of the third law must be considered. Up to the time 
 of Newton no strict inquiry had been made as to the source
 
 34 MECHANICS. 
 
 or origin of force. The physical universe was supposed to 
 consist of matter and of forces, of which the origin was un- 
 known and did not need to be known. The subject of study 
 was the motion of a body under the action of given forces. 
 Newton perceived, and embodied his perception in the third 
 law, that a force which acts on a body always arises from the 
 action of some other body, and further that this other body 
 is also acted on by a force arising" from the first body. Thus 
 all action on a body arises from an interaction between it and 
 other bodies, and the elements which constitute this inter- 
 action, that 'is, the forces which act on the two bodies, are 
 equal and oppositely directed. 
 
 This being the case, it becomes possible for us to compare 
 bodies with each other with respect to the magnitude of the 
 proportional factors which characterize them. For when two 
 bodies interact, the product ma will be the same for each,, 
 since the forces which act on them are equal in magnitude. 
 If, then, any one body is taken as a standard, it may be made 
 to interact with other bodies, and the value of m for each of 
 these bodies may be determined in terms of the m of the 
 standard body. Experiment shows that the ratio of the two 
 values of m obtained for two bodies will be the same, whatever 
 body be taken as the standard. The quantity m thus obtained 
 for any body is called the mass of the body. It may be denned 
 as the quantity of matter of the body, considered with respect 
 to the effect produced upon it by the action of a force. 
 
 25. Mass. The body whose mass is taken as a standard or 
 unit of mass is a certain piece of platinum, called the standard 
 kilogramme. The standard pound, or the mass of a certain 
 other standard piece of platinum, is also often used as a unit 
 of mass by the English-speaking nations. 
 
 It may be shown by experiment that equal masses, what- 
 ever the material of which they are composed, have equal 
 weights, and it is therefore most convenient to construct sets 
 of standard masses and their multiples and submultiples by 
 weighing them. In physical work, the unit mass which is most 
 commonly employed is the one-thousandth part of the stan-
 
 MECHANICS. 35 
 
 dard kilogramme. This mass is called the gramme. By the 
 aid of the unit mass and of the units of length and time 
 already defined, units may be constructed of all other quan- 
 tities which need to be measured in mechanics. These units 
 are called absolute units. 
 
 Jn particular, the absolute unit of force in the C. G. S. 
 system is that force which will impart to a gramme one unit 
 of acceleration, that is, it is that force which, acting uniformly 
 for one second, will impart to a gramme a velocity of one 
 centimetre per second. This unit of force is called the dyne. 
 The force which will impart to a pound in one second a veloc- 
 ity of one foot per second is also taken as a unit force. It is 
 called the poundal. 
 
 In practice, especially in engineering practice and on a 
 large scale, forces are often measured in terms of what is 
 called the gravitation unit. This unit is the weight of a stan- 
 dard mass. This standard mass may be the gramme, but is 
 more often the kilogramme, the pound, or the ton. The 
 numerical value of a force in terms of the gravitation unit is 
 its ratio to the weight of the standard mass. Since the weight 
 of a body is different at different places on the earth, the 
 gravitation unit is not everywhere the same. It differs, how- 
 ever, so little in different places, that its variations may fre- 
 quently be neglected in practice. 
 
 The name of the gravitation unit is the name of the mass 
 whose weight is taken as the unit. Thus, we speak of the 
 weight of a kilogramme, or simply of a kilogramme, as a unit 
 force. 
 
 Since the weight of a body in absolute units is equal to 
 the mass of the body multiplied by the acceleration which will 
 be imparted to it by its weight, or since Weight=mg, the 
 gravitation unit, that is, the weight of the unit of mass, is 
 equal to g absolute units of force. The value of g at latitude 
 40 is abo"t 980, when the centimetre and the second are the 
 units of length and time. The weight of a gramme is there- 
 fore about 980 dynes. When the foot and second are taken is 
 the units of length and time the value of g is roughly equal to 
 32. The weight of a pound is therefore equal to 32 poundals.
 
 36 MECHANICS. 
 
 26. Units of Work. The work of a force has been defined 
 as the product of the force and of the component of the dis- 
 tance through which its point of application moves in the 
 direction of the force. If the direction of the force and the 
 direction in which its point of application moves are inclined 
 to each other by the angle a, the work of the force is expressed 
 by the formula Fscosa, in which F represents the force and 
 s the distance through which its point of application moves. 
 As may easily be seen from this formula, the work of the 
 force may also be measured by the product of the distance 
 through which the point of application moves and the com- 
 ponent of the force in the direction of motion. According 
 to the value of the angle a, the work of the force may hav 
 any value between Fs and Fs. When the force and the 
 motion of its point of application are perpendicular to each 
 other, the work of the force is zero. When the angle a is less 
 than 90, so that cos o is positive, the work of the force is 
 positive. In this case work is said to be done by the force. 
 When the angle a is greater than 90, so that cos a is negative, 
 the work of the fcrce is negative. In this case we may say 
 either that negative work is done by the force or that work is 
 done against the force. 
 
 The absolute unit of work is the work done by unit force 
 ,"when its point of application moves through unit distance in 
 the direction of the force. The work done by a dyne when its 
 point of application moves in its direction through one centi- 
 metre is the unit of work in the C. G. S. system. It is called 
 an erg. 
 
 In practice it is often found convenient to use as a unit of 
 work the work done by the gravitation unit of force when its 
 point of application moves vertically through a certain dis- 
 tance chosen as unit distance. Thus the work done by a kilo- 
 gramme falling vertically through one metre is often taken as 
 a unit of work, it is called the kilogramme-metre. The foot- 
 pound, or the work done by a pound falling vertically through 
 one foot, is also a unit of work which is often employed by 
 the English-speaking nations.
 
 MECHANICS. 37 
 
 27. Impulse. If a constant force F acts upon a body of 
 mass m for the time t, it will impart to it the velocity v. 
 Since the body in this case has a constant acceleration equal 
 
 to*, the relation which connects these quantities is Ft = mv. 
 
 The product Ft is called the impulse. It is of no special im- 
 portance when the force which acts on the body is constant, or 
 when the force acts according to a known law. But it is 
 sometimes of considerable importance when the force acts ir- 
 regularly or according to no known law, and especially when 
 it acts for a very short time. In such cases it may not be 
 possible to determine the force which is acting at any instant, 
 but the impulse may be determined by observing the change 
 in velocity of the body to which the force is applied, and thus 
 an average value of the force during the time during which it 
 acts may be obtained. 
 
 28. Momentum. The product of the m>is of a body and 
 its velocity is called its momentum. The formula of the last 
 section states that the impulse applied to a body is equal to 
 the change which it causes in the momentum of the body. 
 When a constant force acts on a body, as, for example, when a 
 body falls toward the ground from rest, its momentum ac- 
 quired is proportional to the time during which the force acts. 
 
 When two bodies act on each other with equal and opposite 
 forces, according to the third law, the momenta which they 
 acquire in the same time are eqiial and oppositely directed. 
 
 The momentum of a body was called by Descartes and by 
 Newton the quantity of motion of the body. It measures the 
 quantity of motion in this sense, that if bodies having differ- 
 ent momenta move in directions opposite to equal constant 
 forces which act on them, the times which elapse before the 
 bodies come to rest are proportional to the momenta. 
 
 29. Kinetic Energy. When a constant force acts on a 
 body~ and moves it from rest through any distance, it doea 
 work equal to the product of the force and the distance. In 
 this case, where there are no counteracting forces, the force is 
 equal to the product of the mass of the body and the accelera- 
 
 206520
 
 MECHANICS. 
 
 tion imparted to it. From the last paragraph of 18, the 
 velocity acquired by a body moving over a distance s with the 
 acceleration a is given by the formula v z =Zas. Multiplying by 
 m, and writing ma=F, we obtain the formula Fs=$mv 2 , which 
 expresses the relation between the work done by a force upon 
 a free body and the velocity which the body will acquire. The 
 quantity expressed by %mv* is called the kinetic energy of the 
 body. It is manifestly not a vector, that is, it does not de- 
 pend in any way on direction, and it has no negative values. 
 By a slight extension of the demonstration by which this re- 
 lation has been obtained, we may show that, when a body 
 has a certain velocity, and therefore a certain kinetic energy, 
 on entering upon the distance s, the kinetic energy which it 
 will have as it leaves that distance differs from that which it 
 had on entering upon it, by an amount equal to the work done 
 upon the body as it traverses that distance. If this work is 
 positive, the kinetic energy increases and the work done is 
 equal to the gain in kinetic energy. If the work is negative, 
 the kinetic energy diminishes and the work done is equal to 
 the loss of kinetic energy. Leibnitz considered that the kinetic 
 energy, or rather, the vis viva of a body, that is, the product 
 of its mass and the square of its velocity, measured its 
 quantity of motion. Quantity of motion is so unintelligible a 
 phrase, that the controversy which arose between him and the 
 followers of Descartes about its true significance could never 
 be brought to a satisfactory conclusion. All that can be said 
 is that just as there is a sense in which momentum may fairly 
 be called quantity of motion so there is a sense in which this 
 phrase may be applied to vis viva. Vis viva is quantity of 
 motion in this sense, that if bodies having different values of 
 vis viva move in directions opposite to equal constant forces 
 which act on them, the distances passed over by the bodies 
 before they come to rest are proportional to the quantities of 
 vis viva of the bodies. 
 
 30. Work Done by Combinations of Forces. When two or 
 more forces act on a body at the same time, and the body is 
 displaced through a certain distance, the work done by all the
 
 MECHANICS. 39 
 
 forces is equal to the work that would be done, for the same 
 displacement, by their resultant. To show this we shall con- 
 sider first the case of two forces so acting that the displace- 
 ment of the body lies in the plane of the forces. In this case 
 it is easy to see, by inspection of a diagram in which the forces 
 and their resultant are represented by the parallelogram construc- 
 tion, and the displacement is drawn from the origin in its proper 
 direction, that the sum of the components of the two forces in 
 the direction of the displacement is equal to the component of 
 their resultant in the same direction ; and hence that the work 
 done by the resultant is equal to the work done by the given 
 forces. 
 
 If the displacement of the body is not in the same plane 
 as the forces, we may resolve it into two components, one of 
 which is in the plane of the forces and the other perpendicular 
 to that plane. This perpendicular component will not be in- 
 volved in the work done by the forces, and the work done by 
 them will depend only upon the component which lies in their 
 plane. To this component the demonstration already given 
 applies, and the proposition already stated holds for a dis- 
 placement in any direction. 
 
 If itfore than two forces act, we may proceed as in the 
 above demonstration with two of the forces, and repeat the 
 demonstration for the resultant of these forces and a third 
 force, and so on until all the forces have been considered. In 
 this way we may demonstrate the general theorem that the 
 work done by any number of forces is equal to the work done 
 by their resultant. 
 
 Jt may be that one of the two forces which are doing work 
 on the body is so directed that the work which it does is nega- 
 tive. In that case, on carrying out the construction in the 
 same way as has already been described, it may be proved that 
 the work done by the force which does positive work is equal 
 to the work done by the other force, taken as positive, added 
 to the work done by the resultant of the two forces. This 
 statement is true in general for combinations of more than 
 two forces.
 
 40 MECHANICS. 
 
 The result of work done on a body by a combination of 
 forces is a change in its kinetic energy. The magnitude of this 
 change is equal to the work done by the resultant and is posi- 
 tive or negative according as the resultant does positive or 
 negative work! If the body starts from rest under the action 
 of the given forces, it will move in the direction of their result- 
 ant, and the kinetic energy of the body will increase. 
 
 The result of this discussion may also be stated by saying 
 that the work done by the forces which do positive work is 
 equal to the work done by the forces which do negative work 
 added to the kinetic energy gained by the body. In the special 
 case in which the forces are in equilibrium, so that there is no 
 resultant, the kinetic energy of the body does not change. In 
 this case there is no work done on the body by the system of 
 forces, considered as a whole, and the work done by the forces 
 which do positive work is numerically equal to the work done 
 by those which do negative work. 
 
 31. Potential Energy. There are very many cases in nature 
 in which the forces which act on a body, or at least those 
 forces which are considered in the study of the body's motion, 
 depend only on the position of the body relative to other 
 bodies which act on it. Such forces may, for the present, be 
 called positional forces. In case the bodies upon which they 
 depend remain fixed, the positional forces which act upon a 
 moving body as it traverses any path, will be the same in 
 whichever direction it is moving, and will not depend upon 
 any peculiarities of its motion. The work which is done on 
 the body by those forces, when it traverses any path, is there- 
 fore numerically equal and of opposite sign to the work which 
 is done upon it when it traverses the same path in the oppo- 
 site direction. 
 
 When a body is displaced in such a way that negative work 
 is done upon it by positional forces, it is said to have acquired 
 potential energy. The potential energy thus acquired is meas- 
 ured by the work done against these positional forces, or by 
 the work which they will do if the body is abandoned to their 
 action and retraces its path in the opposite direction.
 
 MECHANICS. 41 
 
 Other forces often act upon a body which do not depend 
 solely on its position. Indeed, in actual mechanical opera- 
 tions, such forces always arise, though they are often ignored 
 in formal problems. When they exist, it is found that they 
 always depend upon the motion of the body. They may there- 
 fore, for the present, be called motional forces. A peculiarity 
 which all motional forces possess in common is that they are 
 always directed oppositely to the direction of the body's 
 motion, so that the work which they do is negative. Work 
 done against such forces does not give to the body potential 
 energy, for the body, if left to itself, will not be set in motion 
 or be made to retrace its path by these forces. 
 
 These positional and motional forces are commonly called 
 conservative and non-conservative forces respectively. The 
 reason for these names will be seen at once if we 'consider 
 these forces with respect to the way in which they affect the 
 mechanical energy of the body upon which they act. If the 
 only forces which are acting are positional, the work done by 
 those forces which do positive work is equal to the sum of the 
 potential energy and the kinetic energy gained by the body. 
 In the special case in which no positive work is done, the sum 
 of the kinetic energy and the potential energy of the body 
 remains constant, or is conserved. If some of the forces which 
 act on the body are motional, the work done against them 
 has no equivalent in potential energy gained by the body. In 
 this case, therefore, if there are no forces doing positive work, 
 the sum of the potential energy and the kinetic energy of a 
 body does not remain constant, or is not conserved. In all 
 cases in which motional forces occur, the sum of the potential 
 energy and kinetic energy of the body continually diminishes. 
 
 32. Uniform Motion in a Circle. A body which is moving 
 in a circle in such a way as to pass over equal distances in 
 equal times does not have a constant velocity, according to the 
 definition of constant velocity in 16. The numerical value of 
 its velocity is always the same, but its direction is continually 
 changing. Therefore, according to the definition of accelera- 
 tion in 17, such a body has an acceleration, and from the
 
 42 MKCHA.NIO. 
 
 general relation between force and acceleration stated in 24, 
 it must be acted on by a force. The problem of determining 
 the way in which this force depends on the vek>city of the 
 body and the radius of the circle in which it moves was first 
 solved by Huygens. We shall investigate it by determining 
 the acceleration of a point moving in the circle as the body 
 moves. The force which acts on the body is then equal to its 
 mass, multiplied by this acceleration. 
 
 Suppose the moving point to pass over the small distance 
 AB in the small time t. We may consider this small arc AB 
 equal to its chord, and we may resolve it into two components, 
 - */. one of which is tangent to the circle at the point A, the other, 
 AD, perpendicular to the tangent at the same point. If the 
 point had no acceleration, it would move from the point A 
 along the tangent to the circle at A and its displacement in 
 the small time t would be represented to a first approximation, 
 and in the limit, as the time t approaches zero, would be repre- 
 sented exactly, by the tangential component. Because of its 
 acceleration, however, it traverses in the time t the distance 
 AD along the radius of the circle toward its centre. We may 
 consider that the acceleration of the point, with which it 
 traverses this distance, is constant when the time t is 'small. 
 The distance AD=s is therefore .given by the formula s=$at 2 . 
 The triangles ABD and ACS are similar, and hence we have, 
 from the proportion among their sides, * = __, from which we 
 obtain s = |!. Substituting this value in the other formula 
 
 for s, we obtain a = Jl_ . Now c is the distance traversed bv 
 Pr * 
 
 the moving point in the time t, and H is the velocity of the 
 
 moving point. We therefore obtain finally for the acceleration 
 of the point, which is directed along the radius toward the 
 centre, the value a = ^L The force which acts on the body 
 
 toward the centre is given by _. This force may arise from
 
 MKCHANICS. 43 
 
 the action of a body placed at the centre, from the tension in 
 a cord or other similar body joining the moving body with the 
 centre, from the pressure of a circular wall, or in other ways. 
 As it is always directed toward the centre, it is called a cen- 
 tripetal force. 
 
 From the equality which we have established between a 
 force and the product of the mass upon which it acts and the 
 acceleration caused by it, it is evident that if we substitute 
 for this product a force equal to it in magnitude, and opposite 
 in direction to the acceleration, we shall have a system of two 
 forces in equilibrium. Applying this method to the case under 
 consideration, the fictitious force which we thus introduce is 
 equal to the centripetal force and is directed away from the 
 centre of the circle. It is equal to, and in the same direction 
 as, the real force or reaction which is exerted upon a body at 
 the centre of the circle, when the centripetal force is due to 
 the action of such a body. The observation of this reaction, 
 which is a real force, has led to a confusion between it and 
 the fictitious force which is equal to it, and it is commonly 
 believed that the moving body is acted on by a force tending 
 to carry it away from the centre. This supposititious force is 
 called the centrifugal force. Strictly speaking, there is no 
 such force as a centrifugal force, but the term is often a con- 
 venient one if it is understood to mean simply the fictitious 
 force which may be substituted for the product of the mass 
 of the body and its acceleration. 
 
 33. Centre of Mass. The existence of a particular point in 
 a body, which can be determined from the weights and the 
 relative positions of the parts of the body, and which is called 
 the centre of gravity, has been explained in 7. When the 
 notion of mass was clearly distinguished from that of weight, 
 it became evident that a point might be defined which could 
 be called the centre of mass of the body. The definition of 
 this point is analogous to that of the centre of gravity. 
 
 We may define the centre of mass of two particles, as the 
 point on the line which joins them, which divides that line 
 into segments inversely proportional to the masses at their
 
 44 MECHANICS. 
 
 extremities. The centre of mass of three particles is obtained 
 by first locating the centre of mass of two of them, supposing 
 a mass equal to the sum of the two masses placed at that 
 point, and then determining the centre of mass between this 
 imaginary mass and the third particle. The centre of mass of 
 a larger number of particles is obtained by an extension of 
 the process just described. 
 
 \Ve may define the centre of mass otherwise by supposing 
 the body or system of bodies to be composed of particles of 
 equal mass. On this supposition, the centre of mass is the 
 point whose distances from the three coordinate planes are 
 equal to the average distances of these equal masses from the 
 same planes. This definition is equivalent to the one first 
 given. Analysis shows that the centre of mass thus deter- 
 mined is a definite point, which is independent of the order in 
 which the masses are combined, in carrying out the first defini- 
 tion, and of the position of the coordinate planes, in carrying 
 out the second definition. 
 
 It is evident that this definition applies equally well to 
 scattered particles, or to a collection of particles constituting 
 an extended body. It is determined solely by the masses and 
 their relative positions, and is consequently independent of 
 the circumstances in which the masses are placed. In this 
 respect it differs from the centre of gravity. For a small body, 
 the weights of its parts are parallel forces, and are propor- 
 tional to the masses of the parts, so that the definition of the 
 centre of mass, when properly modified, is also the definition of 
 the centre of gravity of such a body. The centre of mass and the 
 centre of gravity coincide. But for very large bodies, so extended 
 that the weights of their parts are not parallel forces, there will be 
 no true centre of gravity, except in special cases. For such bodies, 
 the centre of gravity will depend on the position of the body. 
 Even in these bodies,, however, the centre of mass is a definite point. 
 
 The centre of mass is of importance because it is possible 
 to describe the motion of a body or of a system of bodies, in 
 certain important respects, in terms of the motion of the
 
 MECHANICS. 45 
 
 centre of mass. The principal relations which illustrate this 
 statement are the following: 
 
 The velocity of the centre of mass is equal to the resultant 
 of the momenta of the different masses of the system, divided 
 by the sum of those masses. This is equivalent to saying that 
 the momentum of a system of masses in any direction is equal 
 to the momentum in that direction of a mass, equal to the 
 sum of the masses of the system, moving with the velocity of 
 the centre of mass. 
 
 The acceleration of the centre of mass is equal to the re- 
 sultant of the forces which act on the masses of the system, 
 divided by the sum of those masses. This is equivalent to 
 saying that the acceleration of the centre of mass is equal to 
 the acceleration which would be imparted to a mass, equal to 
 the sum of the masses, by a single force, equal to the resultant 
 of the forces which act on the system. 
 
 These laws have been tacitly assumed in all the experi- 
 ments in which we have used moving weights to illustrate the 
 simple mechanical laws. We have treated the weight of the 
 body in each case as if it were a single force, and have treated 
 the body as a mass situated at a point. The real forces were 
 of course the weights of the different parts of the body. Their 
 resultant was the force which we used as the weight of the 
 body, and its point of application, whose motion was studied. 
 was the centre of mass. 
 
 The forces which act upon a system of bodies arise either 
 from the action of bodies outside the system, or from the 
 interaction of bodies in the system. Such forces are called 
 external and internal forces respectively. It is of considerable 
 importance to notice that the motion of the centre of mass is 
 not affected by the action of internal forces. For, by New- 
 ton's third law of motion, these internal forces always occur 
 as pairs of equal and opposite forces. The resultant of each 
 such pair of forces is zero, and they therefore contribute 
 nothing to the resultant force from which the motion of the 
 centre of mass is determined. If the centre of mass is at any 
 time at rest, it cannot be set in motion by the internal forces
 
 46 MECHANICS. 
 
 of the system, and will therefore remain at rest unless exter- 
 nal forces act. 
 
 The kinetic energy of a system of masses is equal to the 
 kinetic energy of a mass, equal to the sum of all the masses, 
 moving with the velocity of the centre of mass, added to the 
 kinetic energies of the different masses of the system, moving 
 with their velocities relative to the centre of mass. That is, 
 if we consider the centre of mass as an origin from which the 
 velocities of the parts of the system may be determined, the 
 kinetic energy of the system will be the sum of the kinetic 
 energies of the parts determined from these velocities. If, 
 besides the motions of the parts relative to the centre of mass, 
 the svstem as a whole has a motion relative to some external 
 point, the kinetic energy of the system relative to that point 
 is obtained by adding to the kinetic energy already obtained 
 an additional amount equal to the kinetic energy obtained by 
 supposing all the system concentrated at the centre of mass, 
 and moving with the velocity of the centre of mass relative to 
 the external point. It follows from this statement that the 
 kinetic energy of a system will never be zero unless each part 
 of the system is at rest. The system as a whole may have no 
 momentum even though its parts are moving, provided its 
 centre of mass is at rest, but it will always have kinetic energy 
 unless each of its parts separately is at rest. The reason for 
 this difference lies in this: that momentum is a vector and 
 that consequently two momenta ^n opposite directions can be 
 added so as to be equal to no momentum ; while kinetic energy 
 is not a vector, is always a positive quantity, and BO when 
 two kinetic energies are added together the result is always 
 positive. 
 
 34. Motion of Extended Bodies. In describing the motion 
 of real bodies, it is convenient to limit ourselves to motion in 
 a plane. Motion in three dimensions is so complicated that 
 even its fundamental characteristics cannot be demonstrated 
 by elementary methods. In this kind of motion, the parts of 
 the body move in the same plane or in parallel planes. The 
 formal object of our study will therefore be a plane figure
 
 MECHANICS. ' 47 
 
 moving in its own plane. At some or all of the points of this 
 plane figure \ve suppose masses placed. 
 
 When a plane figure moves so that all its points move in 
 similar and equal paths, its motion is called a translation. A 
 translation is effected when each point of the body undergoes 
 the same displacement. 
 
 When a plane figure moves so that each point of it. describes 
 an arc of a circle around a single point of the plane, which is 
 the common centre of all the circular paths described, its 
 motion is called a rotation. 
 
 Any displacement of a plane figure in a plane may be 
 obtained by the combination of a translation and a rotacion. 
 To show this we select a point, either in the figure or in an 
 extension of the figure, supposed to move with it, as the centre 
 about which rotation is to be effected. This point is trans- 
 ferred by a translation of the figure from its original position 
 to that which it occupies in the position given the figure by 
 the displacement. The figure is then rotated around this 
 point as centre into the position which is given it by the dis- 
 placement. Since a translation in any direction transfers any 
 straight line drawn in the figure to a new position, in which 
 that line is parallel with its original position, it is plain that 
 the rotation involved in this operation will be the same, that 
 is, will turn the straight line through the same angle, what- 
 ever point in the figure is taken as the centre of rotation. 
 The magnitude and the direction of the translation will in gen- 
 eral depend upon the point chosen as the centre of rotation. 
 
 In particular, a point may always be chosen as the centre 
 of rotation such that no translation is needed to effect its dis- 
 place/nent, which can be effected by a simple rotation around 
 that point as centre. The construction by which this point is 
 found fails when the displacement can be effected by a pure 
 translation. 
 
 35. Rotation. In order to describe the rotation of a body 
 and its dependence upon the forces which act on the body, it 
 is convenient at this point to introduce a method of describ- 
 ing the motion of the body in terms of angular magnitudes.
 
 48 , MECHANICS. 
 
 Consider a plane figure free to rotate about a point. Draw 
 a line from this point to any definite point of the figure. This 
 line merely marks a row of points of the figure. From the 
 same point or centre of rotation, draw another line of indefi- 
 nite length, which is fixed in the plane in space and serves as 
 an axis of reference for the measurement of angles. The posi- 
 tion of the figure in the plane is then determined, at any 
 instant, by the angle between this axis and the line first 
 drawn in the figure. When rotation takes place, the angle be- 
 tween these lines changes, or the figure, as determined by t.he 
 line drawn in it, rotates through a certain angle. This angle 
 is called the angular displacement. Each of the points of the 
 figure moves through an arc of a circle whose radius is its 
 distance from the centre of rotation. If the angle in which 
 the angular displacement is given is measured in radians, the 
 distance traversed by each point of the figure, in the circular 
 arc which it describes, is equal to its distance from the centre 
 multiplied by the angular displacement. 
 
 If the figure is rotating uniformly, so that its angular dis- 
 placements in equal times are equal, it has a constant angular 
 velocity, measured by the angular displacement which occurs 
 in a given time, divided by that time. If the angular displace- 
 ment is not the same for any equal intervals of time, the angu- 
 lar velocity of the figure is variable. Its value at any instant 
 is the limit of the ratio of the angular displacement which 
 occurs to the time in which it occurs, as the time and so also 
 the angular displacement approach zero. From the relation 
 already described between the distance actually traversed by 
 a point in the figure and the angular displacement, it is plain 
 that the velocity of any point is equal to the angular velocity 
 of the figure multiplied by the distance of that point from the 
 centre of rotation. 
 
 When the angular velocity of the figure changes at a uni- 
 form rate, so that equal changes of angular velocity occur in 
 equal times, the figure is said to have a constant angular accel- 
 eration. This constant angular acceleration is the ratio of 
 the change in angular velocity to the time in which that
 
 MECHANICS. 49 
 
 change occurs. If we consider the motion of any point of the 
 body, it is plain that the rate at which the numerical value of 
 its velocity changes is equal to the angular acceleration mul- 
 tiplied by the distance of that point from the centre of rota- 
 tion. 
 
 When we consider a body in rotation with constant angular 
 acceleration a, the relation between the time t which has 
 elapsed since the body began to rotate and the angular velocity 
 u acquired in that time, is given by the formula u=at. From 
 the relation which has been stated between the rate of change 
 of the numerical value of the velocity of a point of the figure 
 and the angular acceleration, which will make the distance 
 passed over by any point in the arc of the circle in which it 
 moves the same as that which a point would pass over in a 
 straight line if it were moving in that line with the ac- 
 celeration ra, we obtain fur that distance s = r(f> = rat 3 , 
 from which we deduce the equation <f> \a.P between the 
 angular displacement and the time t in which it has occurred. 
 From these equations, by eliminating t, we obtain the addi- 
 tional equation 2 = 2a0. These relations, expressed in 
 terms of the angular magnitudes and the elapsed time, are 
 analogous to those obtained for the motion of a body moving 
 in a straight line with a constant acceleration. 
 
 36. Kinetic Energy of a, Rotating Body. A plane figure 
 rotating around a point, if its points are endowed with mass, 
 is similar to a body rotating around an axis passing through 
 the centre of rotation and perpendicular to the plane of the 
 figure. If such a body rotates with a constant angular 
 velocity u, it posesses kinetic energy. The magnitude of its 
 kinetic energy is the sum of the kinetic energies of the masses 
 which constitute it. The velocity of one of these masses, whose 
 distance from the centre of rotation is r, is ?-w, and its kinetic 
 energy is ^mr^u 1 . By adding together the expressions such as 
 this for the kinetic energies of the different masses which make 
 up the body, we obtain for the kinetic energy of the whole body 
 
 ^ ~2i mr*. The factor 2 mr*, which manifestly depends only on
 
 50 MECHANICS. 
 
 the masses which make up the body and on their respective 
 distances from the axis of rotation, is called the moment of 
 inertia of the body. With respect to angular motions of the 
 body, the moment of inertia plays the part of the mass of the 
 body with respect to linear motions. 
 
 37. Effect of Force on a Rotating Body. Moment of Force. 
 When a body free to rotate is acted on by a force, it will in 
 general have an angular acceleration. The relation between 
 the force and the angular acceleration which it imparts to the 
 body may be found, if we study a simple case, from the gen- 
 eral relation, established in 29, between the work of a force 
 and the kinetic energy produced by it. As an example of this 
 simple case, we may consider a heavy wheel, whose moment 
 of inertia is represented by /, mounted on a cylindrical axle 
 whose radius is p, and free to rotate about the line which is 
 the axis of this cylinder. If one end of a flexible cord is at- 
 tached to the axle, and if the cord is then wound several times 
 around the axle, a weight hung on the free end of the cord will 
 be a force so applied that it will set the wheel in rotation. 
 Furthermore, as the cord unwinds, this force will always be 
 similarly applied to the. axle. As the weight F falls it does 
 work, the amount of which is given by Fs. The distance .s 
 through which the weight moves is equal to the angular dis- 
 placement of the axle multiplied by its radius p. The 
 kinetic energy acquired by the wheel during this displacement 
 
 is/ w _ or, from the equation of $3">, is Ia<f>. This kinetic 
 
 energy we may set equal to the work done by the force. When 
 this is done, we obtain the relation F/) = Ia. The product Fp 
 is the moment of force, as already defined in 5. The relation 
 expressed by this equation may therefore be stated by saying, 
 that the moment of force which acts on a rotating body is equal 
 to the moment of inertia of the body multiplied by the angular 
 acceleration imparted by the force. This relation is analogous 
 to the fundamental relation connecting force, mass, and accel- 
 eration in linear motion.
 
 MECHANICS. 51 
 
 If several forces act on a rotating body at once, each of 
 them will impart an angular acceleration proportional to its 
 moment. These angular accelerations may be such as either 
 to increase or to decrease the angular velocity of the body. 
 Those which increase the angular velocity of the body may be 
 considered positive, the others negative. The algebraic sum of 
 all these angular accelerations will be the angular acceleration 
 of the body caused by all the forces. If we consider those 
 moments of force positive which tend to turn the body in one 
 sense, and those negative which tend to turn it in the op- 
 posite sense, we obtain the relation S Fp = la, by adding the 
 moments of force algebraically. The algebraic sum S Fp of 
 the moments of force is equivalent to a single moment of force. 
 
 If the angular acceleration of the body is zero, so that the 
 body is in equilibrium, we obtain 2 F/> .0 as the condition of 
 equilibrium. This condition is the same as that found by ex- 
 periment in 5. 
 
 38. Couple. Moment of Couple. A combination of two 
 forces which are equal in magnitude and opposite in direction, 
 though not acting in the same line, is called a couple. When 
 a couple is applied to a body which is free to rotate about an 
 axis, the moment of one of these forces is always greater than 
 that of the other, since the distance of one of them from the 
 centre is greater than that of the other. The algebraic sum 
 of their moments is equal to the product of either one of them 
 and the distance between the lines in which they act. Its sign 
 is positive or negative according as the more distant of the 
 two forces has the positive or negative moment. 
 
 A couple has no resultant, and therefore no single force 
 can be applied to the body which will counteract the effect of 
 a couple. A couple produces rotation, and the angular accel- 
 eration imparted by it is proportional to the moment of the 
 couple. Since the moment of couple does not depend in any 
 way on the positions of the points of application of the forces 
 which constitute it, but only on the magnitude of those forces 
 and the distance between the lines in which they act, the same 
 couple will produce the same effect in a body at whatever
 
 52 MECHANICS. 
 
 points its forces are applied. And further, two different 
 couples will produce the same effect provided their moments 
 are the same. 
 
 A couple applied to a body free to move in a plane will 
 produce rotation around the centre of mass. For, from the 
 proposition stated in 33, the acceleration of the centre of 
 mass depends upon the resultant of the forces which act on the 
 body. Since the couple has no resultant, the centre of mass 
 will have no acceleration, and the motion of the body will 
 therefore be a rotation around the centre of mass. 
 
 39. Effect of a Force on a Free Body. When a force is ap- 
 plied to a free body, its effect will be, in general, to impart 
 acceleration to the centre of mass and also to cause rotation 
 around that centre, This may be shown for the plane figure 
 as follows: Apply to the centre of mass two forces each 
 numerically equal to the given force, one of them parallel to 
 the given force and in the same direction, the other opposite 
 to it. These two forces being equal and opposite and applied 
 at the same point, will have no effect on the motion of the 
 body. They constitute, along with the given force, a system 
 of three forces. According to the proposition of 33 the appli- 
 cation of the given force to the body will impart an accelera- 
 tion to the centre of mass. This acceleration may be con- 
 sidered as arising from the action of that one of the two 
 forces which is applied to the centre of mass and is parallel to 
 and in the same direction as the given force. The other two 
 forces may be considered as constituting a couple which will 
 produce rotation around the centre of mass. 
 
 40. The Physical Pendulum. Any body which is free to 
 rotate about a fixed horizontal axis and which swings back and 
 forth in a vertical plane under the action of its own weight, is 
 a physical pendulum. As was explained in 23, the time of 
 oscillation of such -a pendulum must be the same as that of a 
 certain simple pendulum. The problem before us is to deter- 
 mine, from the characteristics of the physical pendulum, the 
 length of the simple pendulum whose period of oscillation is 
 the same as that of the physical pendulum. When the pendu-
 
 MECHANICS. 53 
 
 lum is not swinging, the line drawn from the axis of sus- 
 pension to its centre of gravity is vertical. When the pendu- 
 dum swings, the extent of its swing is measured by the angle 
 <f> which this line drawn in the pendulum makes with the 
 vertical. The moment of force which acts upon the pendulum 
 when its deviation is <j> is its weight Mg applied at its centre 
 of gravity multiplied by the distance R sin< from the axis 
 of suspension to the line of direction of the weight. This be- 
 ing so, the angular acceleration of the pendulum varies with 
 <f> and is given at the instant at which its deviation is by 
 the formula Mg.Rsin 0=/o. The motion of the pendulum 
 is therefore such that its angular acceleration is proportional 
 
 to the ijine of its deviation, and the factor M 9 R \ s tne factor 
 
 of proportion. 
 
 Considering now the case of the simple pendulum, we see 
 that the acceleration of the bob, tangent to its circular path, 
 is the component of the acceleration g in that direction. This 
 component is given by g sin <f>. The acceleration is equal to 
 the angular acceleration of the pendulum multiplied by its 
 length r. Introducing this value for it, we obtain for the 
 
 angular acceleration the formula a = ?. sin <j>. A comparison 
 
 of this formula with the one obtained for the physical pendu- 
 lum in the preceding paragraph shows that when the deviations 
 of the two pendulums are the same, that is, when the values of 
 <(> are. the same in both formulae, the accelerations will be the 
 same and therefore the motions will be in every respect the 
 
 same, if the factor of proportion ff is equal to the corre- 
 sponding factor of proportion . Setting these two quantities 
 
 T. 
 
 equal, we obtain for the length of the simple pendulum which 
 will swing in the same period as the physical pendulum the 
 
 expression r . The quantity 7 is the moment of inertia 
 
 MR
 
 51 MECHANICS. 
 
 of the pendulum about its axis of suspension. The quantity 
 AIR is called the static moment. 
 
 The problem of finding the period of oscillation of the 
 physical pendulum is thus reduced to the problem of finding 
 the period of the equivalent simple pendulum. This problem 
 cannot be solved by elementary methods in the general case, 
 when no limit is placed on the deviation. We shall therefore 
 consider only the special case in which the deviation is always 
 so small that the arc expressing it may be substituted for its 
 sine. In this case, we may state the condition to which the 
 motion of the pendulum always conforms by saying that its 
 angular acceleration is everywhere proportional to its angular 
 displacement. If we multiply both sides of the equation 
 
 a = ^ rf>, expressing this relation, by r, we obtain on the left 
 
 r 
 the acceleration of the pendulum bob in its path, and on the 
 
 right the factor of proportion *? multiplied by the displacement 
 
 s = r <f> of the bob from the centre of its path. The acceleration 
 of the bob is therefore proportional to its displacement and 
 
 the factor of proportion is ?. 
 
 By the following construction we may describe a motion 
 which possesses this characteristic relation between the ac- 
 celeration and the displacement, and from it we may determine 
 the period of the pendulum. Draw the straight line LOK to 
 represent the path of the pendulum bob. Upon it as diameter 
 construct a circle. Take any point A in this circle and let 
 fall the perpendicular AB upon the diameter. If the point A 
 moves with the numerically constant velocity v around this 
 circle, we may prove that the motion of the point B, which is 
 the point of intersection of the diameter LK, and the perpen- 
 dicular let fall upon it from the moving point A, will be such 
 that its acceleration will be proportional to its distance OB 
 from the centre of the circle, or to its displacement. For, the 
 acceleration of the point B is the component along the diameter 
 of the acceleration of the point A. The acceleration of the
 
 MECHANICS. 55 
 
 point A is directed along the radius OA toward the centre and 
 
 is equal to * , if a is the length of the radius. Its component 
 a 
 
 parallel with the diameter has the same ratio to the accelera- 
 tion of the point A as the displacement OB=s has to the 
 
 radius a, or is equal to . -. Comparing the motion of the 
 a a 
 
 point B with that of the pendulum bob, we see that when the 
 displacements are the same, the accelerations will be the same 
 
 at each point when the factors of proportion !L and ? are 
 
 equal. And also, in this case, the period in which the pendu- 
 lum bob describes its path is the same as the period of the 
 point B. Now, on the suppositions by means of which the 
 motion of the point B has been described, we may determine 
 the period T of the point B. It is evidently the time taken by 
 the point A to describe the circle. The velocity of the point A 
 is the ratio of the circumference to the period and is given by 
 
 v= _?[?. Substituting this value of v in the equation V = ?> 
 we obtain, for the value of the period, T= 2v It. This is the 
 
 be 
 
 time required by the simple pendulum to execute a double 
 vibration. The time required to execute a single vibration, 
 
 which is that commonly observed, is t ir IL. It is to 
 
 W0 
 
 noticed that this value of the period is independent of the dis- 
 placement of the pendulum, and will therefore be the same 
 for any displacement within the limits assumed at the outset 
 of the discussion. 
 
 The point in the physical pendulum whose distance, meas- 
 ured along the line drawn from the point of suspension 
 through the centre of gravity, is equal to the length of the 
 simple pendulum which will swing in the same period, is called 
 the centre of oscillation of the pendulum. It possesses an- 
 other peculiarity, in consequence of which it may also be called 
 the centre of percussion. If an impulse is applied horizon-
 
 56 MECHANICS. 
 
 tally to it when the pendulum is at rest, the motion which 
 occurs is a pure rotation about the axis of suspension, or 
 brings no strain on the axis. Furthermore, if the pendulum 
 is reversed and made to swing about an axis passing through 
 the centre of oscillation, its period will be the same as before. 
 A pendulum so adjusted that its periods of oscillation about 
 two points of suspension, both of which lie on a line passing 
 through the centre of gravity, are equal is called a reversible 
 pendulum. The distance between these points of suspension 
 is the length of the simple pendulum which has the same 
 period. 
 
 41. Motion in Three Dimensions. The motion of a body in 
 three dimensions may be described in a way which is in some 
 respects analogous to that employed for the description of the 
 motion of a plane figure. Any displacement of a body may be 
 accomplished by a translation and a rotation around a suit- 
 ably chosen axis. A direction of translation may always be 
 found such that the axis around which the necessary rotation 
 takes place is in the same direction. An infinitesimal dis- 
 placement of the body may thus be analyzed into an infinites- 
 imal translation and an infinitesimal rotation around an axis 
 drawn in the same direction as the translation. This motion 
 is that of a part of a screw, when the screw is turned and so 
 driven forward, and the motion of a body at any instant may 
 therefore be called a screw motion. As the body moves 
 through a finite path, the characteristics of the screw whose 
 motion describes the motion of the body, that is, the pitch of 
 the screw and the direction of its axis, change from instant 
 to instant. The changes, however, are never discontinuous. 
 
 The study of the moment of inertia of a body around an 
 axis passing through the centre of mass shows that this 
 moment of inertia can always be found if we know the mo- 
 ments of inertia around three principal axes which pass 
 through the centre of mass and are perpendicular to each 
 other. Around one of these axes the moment of inertia has 
 a maximum value, around one of the others a minimum value. 
 When the body is rotating around its axis of greatest moment
 
 MECHANICS. 57 
 
 of inertia, or around its axis of least moment of inertia, it is 
 in a condition of kinetic stability, that is, an impulse applied 
 to the body, though it will compel the body to rotate around 
 a new axis, will not so alter the motion of the body as to cause 
 the new axis to deviate more and more from the direction of 
 the original axis. If the original rotation is around the third 
 of the principal axes, its condition is unstable, that is, though 
 the rotation will persist around that axis so long as no im- 
 pulse is applied to the body, the application of an impulse will 
 cause a rotation around a new axis, whose direction will con- 
 tinually deviate more and more from that of the .original axis. 
 A rotation set up arond any other axis than one of these 
 three will not continue around that axis, even though no im- 
 pulse is applied to the body, but the axis around which rota- 
 tion occurs will change its direction in the body continually. 
 
 When a body is in rotation about its axis of greatest 
 moment of inertia, the application of a small couple, to pro- 
 duce rotation about a perpendicular axis, will result in such 
 a combination of motions that the final effect is a rotation 
 about a third axis perpendicular to both the others. This 
 statement is illustrated by the instrument called the gyro- 
 scope or gyrostat. Rotation of a body about an axis of great- 
 est moment of inertia thus introduces a resistance to any 
 force which is so applied as to change the direction of that 
 axis. Thus if a heavy wheel is mounted inside a box and is 
 kept in continual rotation, the box, though of itself it does 
 not originate motion, nor resist a motion of translation, will 
 offer a resistance to any force tending to turn it around. Cer- 
 tain forces which exist in nature have been explained by sup- 
 posing that the bodies which exhibit them contain portions 
 W 7 hich are in rapid rotation. Since these rotating portions, 
 even if they exist, are such that they can never be perceived, 
 they are said to be concealed, and the forces exhibited by the 
 bodies are said to be due to concealed motion. 
 
 'The angular velocity of a body rotating around an axis 
 may be conceived of as made up of two angular velocities 
 about other axes passing through this axis at the same point.
 
 58 MECHANICS. 
 
 The magnitudes of these component angular velocities, as they 
 may be called, may be found by laying off on the original 
 axis, from the origin or point at which the three axes inter- 
 sect, a distance which is numerically equal to the angular 
 velocity about that axis, and constructing upon this distance 
 as diagonal the parallelogram whose sides are in the directions 
 of the other axes. This statement may be illustrated by the 
 Foucault pendulum. This pendulum consists of a heavy bob 
 suspended by a long cylindrical wire. The upper end of the 
 wire is held in a clamp in such a way that it may swing with 
 equal freedom in any direction. Arrangements are made for 
 noting the plane in which the pendulum swings at any in- 
 stant. The pendulum thus arranged is set swinging, as ex- 
 actly as possible, in a vertical plane, and the plane of its 
 swing is noted from time to time. It is found that this plane 
 changes its apparent direction with respect to the surface of 
 the earth, and that the angular deviation of the plane of swing 
 from its original position, in a given time, is at different 
 places proportional to the sine of the latitude of the place. 
 We may give a kinematic description of this result as follows: 
 If we conceive such a pendulum set up at the North Pole, the 
 plane of its swing will remain fixed in space and the earth will 
 turn around under it, so that to an observer examining the 
 pendulum, its plane of swing will change its position relative 
 to the earth's surface. In one day it will appear to have 
 traversed a complete circle. If the same pendulum is set up 
 at the .Equator, and is set swinging in the north and south line, 
 the earth will carry it around, and its plane of swing will not 
 change its position relative to the earth. At any intermediate 
 station the motion of the plane of swing may be determined 
 as follows: Draw a line from the centre of the earth to the 
 station and another line perpendicular to this, in the plane 
 containing this line and the earth's axis. We may conceive 
 the angular velocity of the earth resolved into two angular 
 velocities around these two axes. The angular velocity around 
 the perpendicular axis last drawn will have no effect upon the 
 motion of the pendulum. The angular velocity about the other
 
 MECHANICS. "59 
 
 axis, however, which is equal to the angular velocity of the 
 earth multiplied by the sine of the latitude, may be considered 
 as an angular velocity with which the earth turns under the 
 pendulum around the axis passing through the station. The 
 plane of swing will therefore have an apparent angular velocity 
 around this axis which is proportional to the sine of the lati- 
 tude. The full dynamical discussion of the motion of the Fou- 
 cault pendulum cannot be given here.
 
 MECHANICS OF LIQUIDS. 
 
 MECHANICS OF LIQUIDS. 
 
 42. The Problem of the Crown. The study of the peculiar 
 mechanical effect*, exhibited by liquids was "begun by Archi- 
 medes. According to the story told by Vitruvius, a certain 
 quantity of gold had been given by King Hiero to. a goldsmith, 
 with the order to use it in constructing an elaborate crown. 
 The crown when returned weighed as much as the gold which 
 had been supplied, but for some reason Hiero suspected that 
 a part of the gold had been abstracted, and silver substituted 
 for it. He asked Archimedes to devise a way to determine 
 whether this was so or not, without injuring the workmanship 
 of the crown. Archimedes is said to have discovered the way 
 in which this might be done by noticing the way in which the 
 water overflowed from a bath into which he had entered. It 
 is plain that, if bodies of different materials and of equal 
 weights do not occupy equal volumes, the one having the 
 larger volume will cause more water to overflow from a full 
 vessel in which it is immersed than the other one will, and 
 that by determining the overflow caused by the crown and by 
 equal weights of gold and silver, the question asked of Archi- 
 medes might be answered. 
 
 By reflection upon the question thus presented, Archimedes 
 was led to assume certain principles to which a liquid will 
 conform, and, by the aid of these principles, to deduce certain 
 laws which govern the apparent loss of weight of bodies im- 
 mersed in a liquid, and to determine certain cases of equil- 
 iibrium of floating bodies. These principles, which were made 
 by Archimedes the postulates of his theory, were, that when 
 two portions of a liquid are similarly situated and are con- 
 tiguous to each other, the portion which is under less pressure 
 is set in motion by the portion which is under greater pressure; 
 and that the pressure at a point in the liquid is proportional 
 to the height of the column of liquid which stands vertically 
 above it. These principles are not so fundamental as the
 
 MECHANICS OF LIQUIDS. 61 
 
 principle afterwards introduced by Pascal. We shall accord- 
 ingly pass, without further consideration of them, to a state- 
 ment of Pascal's principle and an investigation of its conse- 
 quences. 
 
 43. Pascal's Principle. According to Pascal the character- 
 istic peculiarity of a liquid may be expressed by saying that, 
 when a liquid is under the action of no forces except pressures 
 applied to its external surface, the pressure at every point in 
 it and in every direction around each point is the same. To 
 properly appreciate Pascal's principle, we must clearly define a 
 pressure and understand how it may be measured. To do this 
 in the simplest possible way, let us suppose that we have a 
 cylindrical vessel closed above by a tightly fitting piston, that 
 can move in the cylinder, and that the space below the piston 
 is filled with a liquid. If weights are put upon the piston, 
 their combined effect is a resultant force applied to the piston 
 at a point below their centre of gravity. By a suitable adjust- 
 ment of the weights, this centre of gravity may be made to 
 coincide with the centre of figure of the piston. The piston, 
 with the weights on it, is in equilibrium under the action of 
 this resultant force and of an equal and opposite resultant, 
 which arises from the action upon the piston of the different 
 parts of the liquid in contact with it. Because of the similar- 
 ity of the parts of the liquid to one another, the forces which 
 they exert are thought of as equal and uniformly distributed 
 over the piston. If this be so, the resultant force which is 
 applied to unit area of the piston is found by dividing the 
 total force applied to the piston by its area. This resultant 
 force applied to unit area is called the pressure of the liquid 
 on the piston. Obviously it is possible to obtain the same 
 value for the pressure by supposing the forces applied to the 
 piston to be increased in number and diminished in magnitude 
 without limit, on the condition that the sum of all these 
 forces remains constant and taking the ratio between the re- 
 sultant force which acts on any area and that area. This 
 statement is important in defining pressure at a point, which
 
 52 MECHANICS OF LIQUIDS. 
 
 is the limit of the ratio just stated as the area approaches 
 zero. 
 
 To conceive of a pressure in a liquid we may suppose a 
 plane surface drawn in the liquid and the liquid removed on 
 " one side of it. To keep the surface in equilibrium forces would 
 have to be applied to it, imagining it to be made up of rigid 
 parts. From these forces and the areas to which they are 
 applied, the pressure may be measured. 
 
 A statement which is even more fundamental than Pascal's 
 principle is that the pressure on any surface in a liquid is 
 always normal to it. From this principle Pascal's principle 
 may be deduced. 
 
 An application of Pascal's principle is made in the hydro- 
 static press. This consists essentially of two cylinders joined 
 by a connecting pipe. The diameter of one of these cylinders 
 is considerably larger than that of the other. The cylinders 
 and the pipe connecting them are filled with water, and pis- 
 tons are inserted which rest upon the surfaces of the water 
 in the cylinders. If weights are placed upon the pistons they 
 w T ill be balanced if they are to each other in the ratio of the 
 areas of the pistons on which they stand. The condition of 
 equilibrium which has been stated follows from Pascal's prin- 
 ciple, for from that principle, the pressure on unit area is the 
 same for each piston, and the total pressures on the pistons, 
 or the weights which they will sustain, are therefore propor- 
 tional to their areas. 
 
 44. Pressure in a Liquid Due to Its Weight. When a liquid 
 stands at rest in a vessel, the pressure at any point within it 
 is proportional to the depth of that point below the surface. 
 Let us imagine a circle of unit area placed at the point and 
 parallel with the surface. The cylindrical column of liquid 
 which stands on that area is in equilibrium under the com- 
 bined action of its .weight, of the upward pressure applied to 
 the area on which it stands, and of the pressures applied to its 
 sides. The pressures applied to its sides are perpendicular to 
 the cylindrical surface and exert equal and opposite forces 
 upon this cylinder, so that they do not counteract the weight
 
 MECHANICS OF LIQUIDS. 63 
 
 of the column. The weight is counteracted only by the up- 
 ward pressure applied to the base of the cylinder, and since 
 the weight is proportional to the height of the cylinder, the 
 pressure is also proportional to that height. 
 
 The free surface of a liquid is a horizontal plane. If we 
 describe a cylinder in the liquid around a horizontal line as 
 axis, the pressures acting on its two ends are equal; for, the 
 cylinder is in equilibrium along its axis under these pressures 
 alone, since its weight has no horizontal component. The two 
 points at which the pressures are equal will, from the previous 
 proposition, be at equal distances from the surface, and the 
 surface is therefore horizontal. 
 
 If the mass of liquid is so extended that the weights of its 
 different parts are not parallel, but converge toward the 
 centre of the earth, a slight modification of the demonstration 
 just given will show that the free surface is part of the surface 
 of a sphere, whose centre is the centre of the earth. This 
 form of the proposition was demonstrated by Archimedes. 
 
 From this proposition it may easily be seen that a liquid 
 in two communicating vessels will stand at the same level in 
 both. 
 
 45 Archimedes' Principle. One of Archiinede*' proposi- 
 tions is of such fundamental importance that it is usually 
 recognized as his especial contribution to hydrostatics, and 
 called Archimedes' principle. We may state this principle as 
 follows: When a body is immersed in liquid it loses in ap- 
 parent weight an amount equal to tbe weight of the liquid 
 displaced by it. Of course the body does not really lose weight, 
 but part of its weight is counteracted by the upward pressure 
 of the liquid. The truth of this principle may be seen if we 
 imagine the body removed from the liquid and the space which 
 it occupied filled with the liquid. The whole liquid mass will 
 then be in equilibrium, and the weight of the portion which 
 has been introduced will be equal and opposite to the resultant 
 of the pressures applied to its surface by the surrounding 
 liquid. Exactly these same pressures, having the same re- 
 sultant, were applied to the body when it w 7 as in the liquid,
 
 04 MECHANICS OF LIQUIDS. 
 
 and its apparent weight is therefore less than its real weight 
 bv an amount equal to this resultant force, or to the weight 
 of the liquid which it displaces. 
 
 The method which Archimedes is said to have used in the 
 t actual solution of the problem of the crown will illustrate this 
 principle. The apparatus which he employed was an equal 
 armed lever or balance, on one arm of which a sliding weight 
 could be moved. A weight of gold equal to the weight of the 
 crown was hung on one end of this balance and an equal weight 
 of silver on the other. The masses of gold and silver were next 
 immersed in water. The balance was then no longer in equi- 
 librium, the gold appearing to be heavier than the silver. 
 Equilibrium was restored by placing the sliding weight at a 
 certain distance from the point of suspension. It is plain, 
 from Archimedes' principle, that the moment of force intro- 
 duced by this weight was equal to the moment of force in 
 the opposite sense introduced by the excess of upward pressure 
 exerted by the water on the silver over that exerted on the 
 gold. This moment, therefore, was proportional to the excess 
 of volume of the silver over that of the gold. The crown was 
 next substituted for the silver, and the gold and the crown 
 immersed in water. Since the crown contained silver, its 
 volume was greater than that of the gold, and there was an 
 excess of upward pressure on it. Equilibrium was restored by 
 placing the sliding weight at another distance from the axis 
 of suspension. By reasoning similar to that already employed, 
 we conclude that the moment of force introduced by the weight 
 in this case was proportional to the excess of the volume of 
 the crown over that of the gold. This excess was due to the 
 silver in the crown, and proportional to the amount of silver 
 present, so that the ratio of the quantity of silver in the crown 
 to the quantity of silver which weighed as much as the crown, 
 was equal to the ratio of the moments of force observed in the 
 two experiments, or to the ratio of the arms to which the 
 sliding weight was applied. 
 
 46. Floating Bodies. When a body is so large and weighs 
 so little that it displaces, when completely submerged, a weight
 
 MECHANICS OF LIQUIDS. 65 
 
 of liquid greater than its own weight, it will not sink in the 
 liquid. It will only be immersed so far that the weight of the 
 water which it displaces is exactly equal to its own weight. 
 It then floats upon the liquid. When such a body is placed 
 upon the surface of the liquid it will generally not remain at 
 rest, but will turn round until it attains some definite position 
 of equilibrium. If it is afterwards slightly disturbed from 
 this position, it will come back to it again if its equilibrium is 
 stable. When it is in equilibrium, its weight applied at the 
 centre of gravity and the resultant of the upward pressures 
 which are applied to its surface must be equal and opposed to 
 each other in the same vertical line. The centre of gravity 
 of the liquid which is displaced by the body is the point at 
 which the resultant of the upward pressures may be considered 
 as applied. The vertical line joining the centre of gravity of 
 the body and the centre of gravity of the displaced liquid is 
 called the axis of the body. If the axis is given an infinitesi- 
 mal inclination, the point at which it is intersected by the 
 resultant of the upward pressures is called the metacentre. 
 The body is in stable equilibrium when the metacentre lies 
 above its centre of gravity. In this case, when the body is 
 inclined, its weight and the resultant of the upward pressures 
 combine to form a couple which turns the body back into its 
 original position. 
 
 47. Specific Gravity or Relative Density .If we weigh 
 equal volumes of diiferent substances we find that in general 
 their weights are different. When the volume which is chosen 
 is some standard volume, these different weights are charac- 
 teristic of, or specify, the different substances. By the aid of 
 Archimedes' principle it is easy to compare the weight of a 
 body with the weight of an equal volume of water, taken as 
 a standard substance. The ratio of the weight of the body to 
 the weight of an equal volume of water is a characteristic 
 number for the substance of which the body is composed. It 
 is therefore called the specific gravity of that substance. 
 
 The mass of a substance which is contained in unit volume 
 is called the density of that substance. From the proportion-
 
 66 MECHANICS OF LIQUIDS. 
 
 ality which exists between mass and weight, the specific grav- 
 ity of a substance is the ratio of its density to the density of 
 the standard substance water. Specific gravity is therefore 
 also called relative density. 
 
 When the metric system of weights and measures was 
 introduced, the attempt was made by Borda to construct the 
 standard mass, or kilogramme, equal to the mass of a cubic 
 decimetre of water under standard conditions. In this at- 
 tempt he succeeded very nearly, so that unless extraordinary 
 accuracy is desired, the mass of a quantity of water in kilo- 
 grammes is numerically equal to the volume of the water in 
 cubic decimetres. The gramme is the mass of water in a cubic 
 centimetre. In the system of units which we employ, in 
 which the gramme is the unit of mass, the centimetre, the unit 
 of length, and the cubic centimetre, the unit of volume, the 
 density, defined as the ratio of the mass to the volume, is 
 expressed by the same number as the relative density. In 
 other systems of units this is generally not the case. The 
 relative density, which is a simple ratio between two densi- 
 ties, is the same for all systems of units; the absolute density 
 is different in the different systems. 
 
 We now proceed to consider some of the methods by which 
 the relative density of a substance may be determined. 
 
 A vessel may be arranged full of water, so that the water 
 which overflows when a body is immersed in it can be caught 
 and weighed. The volume of the water which overflows is 
 equal to the volume of the body, and the ratio of the weight 
 of the body to the weight of the water is therefore the rela- 
 tive density. 
 
 The density of solids which are insoluble in water is com- 
 monly obtained by the use of the hydrostatic balance. This 
 is a balance so constructed that the body to be tested can 
 hang below one of the scale pans. The body is first weighed 
 in air. It is then immersed in water and its apparent weight 
 determined. By Archimedes' principle, the loss in weight 
 which it apparently undergoes is the weight of an equal vol- 
 ume of water. The ratio of the body's weight to this loss in
 
 MECHANICS OF LIQUIDS. 67 
 
 weight is therefore the relative density of the substance of 
 the body. , 
 
 The relative density of a liquid is often determined by an 
 instrument called the hydrometer. Hydrometers are of two 
 kinds, those of constant weight, and those of constant volume. 
 The hydrometer of constant weight is generally a glass bulb 
 weigfiled at the bottom, and furnished above with a long 
 cylindrical stem. When it is placed in a liquid it stands up- 
 light in it, Jind a proper adjustment of its w r eight will cause 
 it to sink so that part of its stem is immersed. It is standard- 
 ized by placing it in water, and marking the point on the stem 
 at which the stem protrudes from the water surface. It is 
 then placed in another liquid of known density, and the place 
 on the stem again marked at which it protrudes from the sur- 
 face of the liquid. The distance between these two points is 
 marked off into any arbitrary number of equal divisions, and 
 this graduation is extended above and below the standard 
 points. When the instrument, thus prepared, is placed in any 
 liquid, it sinks to a certain point, which may be read off by 
 means of the graduation. Its weight remaining always the 
 same, the weight of the liquid which it displaces is always the 
 same, and since the point to which it sinks determines the 
 volume of the liquid displaced, we have the means of determin- 
 ing the volumes of equal weights of different liquids, and so 
 of determining their relative densities. The hydrometer of 
 constant volume is a glass or metal bulb, weighted at the 
 bottom, and carrying at the top a cylindrical stem, supporting 
 a pan in which weights may be placed. The instrument is 
 floated in a liquid and weights are placed in the pan until the 
 instrument sinks to a marked point on the stem. The weight 
 of the instrument and the weights in the pan are together 
 equal to the weight of the liquid displaced. By using this 
 instrument in different liquids, one of which is the standard 
 liquid, we may determine the weights of equal volumes of the 
 liquids, and therefore their, relative densities. 
 
 The rejative density of liquids is often determined by the 
 use of the specific gravity bottle, or pyknometer. This is a
 
 68 MECHANICS OF LIQUIDS. 
 
 small bottle or flask or other vessel of constant volume. The 
 weight of water which will fill it is determined, once for all. 
 To determine the relative density of another liquid, it is filled 
 with that liquid and the weight of the liquid determined. 
 
 48. The Barometer. The attention of Galileo' was once 
 ^called to the fact that the water of a certain deep well could 
 not be drawn out of it by a pump. The water rose in the pipe 
 about 34 feet and could not be raised further by continued 
 pumping. At that time the rise of water in a pump was con- 
 sidered an exhibition of a general principle, embodied in the 
 statement "Nature abhors a vacuum." Galileo recognized that 
 the fact which had been called to his attention proved at least 
 that this principle was limited in its application, but he -was 
 not able to explain it. 
 
 Torricelli, one of his friends, starting with the knowledge 
 which he had obtained from Galileo that air has weight, con- 
 sidered it to be a simple case of equilibrium between the 
 pressure of the water column in the pipe and of the air. To 
 verify this conclusion, he filled a long glass tube, closed at one 
 end, with mercury, and stopping the other end with his finger, 
 inverted the tube and inserted the lower end of it in a vessel* 
 of mercury. On removing his finger, the column of mercury 
 in the tube fell until its highest point was about 30 inches 
 above the surface of the mercury in the vessel and settled to 
 rest in that position. On the supposition that the mercury 
 column is sustained by the pressure of the air on the free 
 surface of the mercury in the vessel, the height of the mercury 
 column ought to be proportioned to the height of the water 
 column, in the experiment with the pump, inversely as the 
 respective masses of mercury and water which occupy the same 
 volume. For, the pressure of the air is the same in both cases, 
 and therefore the pressures of the two columns of mercury 
 and water should be the same, if the hypothesis is correct. 
 The general expression for the pressure of a column of liquid 
 may be found by considering a column of unit cross section 
 and of height h. The volume of such a column is also ex- 
 pressed by h, and the weight of it is equal to its volume, multi-
 
 MECHANICS OF LIQUIDS. 69 
 
 plied by the weight contained in unit volume. This product 
 then measures the pressure. Two liquid columns will exert 
 the same pressure when this product is the same for both of 
 them. Their heights are therefore inversely as the weights of 
 each contained in unit volume. To apply this to the case under 
 consideration we need, in addition to the data already given, 
 the fact that the weight of a given volume of mercury is 13.6 
 times as great as the weight of an equal volume of water. 
 Accordingly, the height of the water column sustained by the 
 pressure of the air should be 13.6 times as great as the height 
 of the corresponding mercury column. This conclusion from 
 Torricelli's hypothesis was verified by his observations, and 
 he accordingly concluded that the water rose in the pump and 
 that the mercury was sustained in the tube by a common 
 cause, the pressure of the air. 
 
 The arrangement constructed by Torricelli may be set up 
 permanently as a means of measuring the pressure of the 
 atmosphere. It is then called a barometer. The pressure of 
 the atmosphere is found to vary from time to time, but it 
 never differs very much at any one place from a mean or 
 standard pressure for that place. 
 
 Pascal contributed to the verification of Torricelli's 
 hypothesis by showing that the height of the mercury column 
 in the barometer diminishes when the barometer is carried up 
 a mountain. The reason for this is plain when we notice that 
 the barometer at the higher station is relieved from the 
 pressure of a column of air, whose height is the difference of 
 level between the lower and upper stations. 
 
 The pressure indicated by the barometer under standard 
 conditions is often used as a unit of pressure. This unit of 
 pressure is called the pressure of one atmosphere, or some- 
 times the atmo. It is the pressure indicated by the barometer 
 when placed at the sea-level and when the height of the column 
 is 760 millimetres. 
 
 49. Flow of Liquids. Torricelli's Theorem. It was ob- 
 served by Torricelli that when a stream of water issues from
 
 70 MECHANICS OF LIQUIDS. 
 
 a small orifice in the side of a vessel full of water, the velocity 
 of the stream is the same as that which a body would acquire 
 by falling from the surface of the water to the orifice. This 
 relation is called Torricelli's theorem. It may be examined by 
 fitting to the orifice a short tube so bent that the stream 
 issuing from it is directed upwards, in which case it will be 
 found that the stream rises to the same level as that of the 
 water in the vessel. It may also be examined by allowing the 
 stream to issue horizontally and determining the dimensions 
 of the parabola which it describes. 
 
 When we attempt another verification of Torricelli's state- 
 ment, by a determination of the quantity of water which 
 issues in a given time, we obtain results inconsistent with it. 
 The amount of water obtained is always less than that which 
 should issue from an orifice of the given size, if the velocity 
 is that stated in Torricelli's theorem. Newton observed that, 
 owing to the way in which the parts of the water near the 
 orifice rush together as they issue from it, the diameter of the 
 stream outside the orifice is not as great as that of the orifice. 
 The narrow portion of the stream is called the vena contracta. 
 When the quantity of water which issues is calculated as if it 
 were issuing from an orifice whose diameter is that of the 
 vena contracta, the results obtained are consistent with the 
 calculation. 
 
 When a liquid fiows through a channel or series of pipes, 
 whose cross-sections are different at different places, the 
 velocity of the liquid will vary along the channel, being at 
 different points inversely as the cross-sections at those points. 
 
 When a columnof liquid is moving along a pipe, the pressure 
 of the liquid outward, against the wall of the pipe, is dimin- 
 ished by an amount which depends upon the velocity of the 
 liquid. This diminution of pressure is theoretically propor- 
 tional to the square of the velocity. The velocity may there- 
 fore be so great that the pressure against the wall of the pipe 
 disappears, in which case the liquid will flow past an orifice 
 made in the side of the pipe, without issuing from it. If the 
 velocity is still further increased, a negative pressure, or one
 
 MECHANICS OF LIQUIDS. 71 
 
 directed inwards, will be set up, the air will enter through the 
 orifice from without and will be carried along with the liquid. 
 
 50. Waves on the Surface of a, Liquid. When a liquid 
 surface is disturbed, the disturbance passes outward in every 
 direction as a wave or a series of waves. Each of these waves 
 consists of a portion of the liquid which is elevated above the 
 general level, and another portion which is depressed below it. 
 Our common observation of the movements of floating bodies 
 when waves pass under them, shows that the liquid is not car- 
 ried along with the waves. The wave is therefore simply a 
 mode of motion which is impressed successively upon the dif- 
 ferent parts of the surface. We may study the motion of the 
 liquid caused by the waves, by immersing in it small fragments 
 of some solid whose relative density is the same as that of the 
 liquid. In the case of water small fragments of amber may be 
 used. If a succession of similar waves passes over the surface, 
 the fragments in the surface describe circles, whose diameters 
 are equal to the vertical distance between the highest point 
 and the lowest point of the wave. At the highest point of 
 the wave they move forward in the direction in which the wave 
 is travelling. At the lowest point they move backward with 
 the same velocity. The fragments which are below the surface 
 describe ellipses, which become smaller and smaller, and more 
 and more excentric as the depth increases, until at last they 
 cannot be distinguished from minute horizontal lines. 
 
 The disturbance thus described does not penetrate far be- 
 low the surface, and when the depth of the liquid is very great, 
 most of it remains quiet, even when the waves which pass over 
 it are large. When the depth of the liquid is not great, the 
 backward part of the motion, which is in the lower part of 
 the curves described by the parts of the liquid, is retarded, 
 and the forward motion is in excess. The top of the wave thu| 
 moves forward faster than the bottom, and if the wave ad- 
 vances in such a direction that the depth continually decreases, 
 it will at last curl over and break.
 
 72 GRAVITATION OB MASS ATTRACTION. 
 
 GRAVITATION OR MASS ATTRACTION. 
 
 51. Kepler's Laws of Planetary Motion. By an extended 
 study of the apparent motions of the planet Mars, and then 
 of those of the other planets, Kepler showed (1609-1619) that 
 they could be represented by supposing the motions of the 
 planet to conform to the following laws, which are known as 
 Kepler's laws: 
 
 1. The path of a planet is an ellipse, at one of the foci of 
 which the sun is situated. 
 
 2. The radius vector drawn from the sun to a planet 
 sweeps out equal areas in equal times. 
 
 3. The squares of the periodic times of the planets are 
 proportional to the cubes of the semi-major axes of their 
 elliptic orbits. 
 
 By the periodic time of a planet is meant the time required 
 by it to completely describe its orbit. 
 
 52. The Law of Gravitation or Mass Attraction. The laws 
 of Kepler merely describe motions which are consistent with 
 the apparent motions of the planets. On the assumption that 
 they describe the real motions of the planets, it became a 
 question of the utmost interest to determine the forces which 
 caused the motions. This question was first answered by New- 
 ton in his Principia (1687). Newton's investigation is an ex- 
 cellent illustration, perhaps the best illustration which we 
 have, of the process of thought employed in physical reasoning, 
 as described in 2. 
 
 Newton first made the hypothesis that each planet is acted 
 on by a force which is proportional to its mass, is directed 
 toward the sun, and is inversely proportional to the square of 
 the distance between the planet and the sun. He then applied 
 the principles of mechanics to determine what would be the 
 motion of a body acted on by a force directed toward a fixed 
 point according to this hypothesis.
 
 GRAVITATION OR MASS ATTRACTION. 73 
 
 He showed that if the force which acts on the body is 
 directed toward a fixed point, whatever may be the way in 
 which the force varies with the distance of the body from 
 that point, the radius vector drawn from the point to the 
 body will sweep out equal areas in equal times. That is, a 
 body acted on by a force which is always directed toward a 
 fixed centre conforms to Kepler's second law. Hence the con- 
 clusion may be drawn that the force which acts on the planet is 
 directed toward the sun. 
 
 Newton showed further that when the force which acts on 
 the body varies inversely with the square of the distance be- 
 tween the body and the fixed centre, and is directed toward 
 that centre, the path of the body will be one of the conic sec- 
 tions. Whether the path of the body is an ellipse, a parabola, 
 or an hyperbola depends on the velocity of the body as it 
 passes the end of the major axis. When this velocity does 
 not exceed a certain limit, the path of the body will be an 
 ellipse. The motion of the body, therefore, which is caused by 
 the hypothetical force supposed to act on it, conforms to Kep- 
 ler's first law. Hence the conclusion may be drawn that the 
 force which acts on a planet toward the sun varies inversely 
 with the square of the distance between the planet and the 
 sun. , 
 
 Newton showed further that when forces directed toward 
 a common centre act on different bodies so as to make them 
 describe elliptic orbits, the squares of the times in which they 
 describe their orbits are proportional to the cubes of the semi- 
 major axes of the orbits, if only the force which acts on each 
 body is proportional to the mass of the body. That is, if the 
 forces which act upon the bodies are proportional to their 
 masses, the motions of the bodie will conform to Kepler's 
 third law. Hence the conclusion may be drawn that the forces 
 which act vipon the planets are proportional to their masses. 
 
 The results of the preceding paragraphs, which may be 
 summed up in the statement that the planets move as if they 
 were attracted toward the sun by forces which are propor- 
 tional to their masses and which vary inversely with the
 
 74 GKAVITATION OK MASS ATTRACTION. 
 
 squares of their distances from the sun, are all that can be 
 reached by the study of the motion of the planets as expressed 
 in Kepler's laws. More might have been learned from a study 
 of the perturbations caused in the orbit of one planet by the 
 action of the others around it. Newton, however, did not 
 proceed alone this line in his argument, but followed another 
 course. He showed that the motion of the moon conforms to 
 Kepler's first and second laws when the earth is taken as the 
 centre of force, and therefore concluded that the moon is acted 
 on by a force directed toward the earth and varying inversely 
 with the square of the distance of the moon from the earth. 
 He then showed, from a calculation of the dimensions of the 
 moon's orbit, that the acceleration of the moon toward the 
 earth is equal to the acceleration of a body near the earth's 
 surface diminished in the inverse ratio of the squares of the 
 distances of the moon and of the body from the centre of the 
 earth. From this the conclusion was drawn that the force 
 whicli acts on the moon and determines its motion in its orbit 
 is of the same nature as the force which causes heavy bodies 
 to fall toward the earth; and further, that the forces which 
 act between the different bodies of the planetary system are 
 of the same nature. This force is therefore called the force 
 of gravitation. 
 
 Newton constructed two pendulums of equal length whose 
 bobs were two similar spherical boxes, in which different 
 bodies could be placed. He found that whatever were the sub- 
 stances placed in the boxes, the times of oscillation of the 
 pendulums were the same, and hence concluded that the force 
 of gravity is proportional exactly to the mass of the body 
 acted on. and is independent of its other characteristics. 
 
 Observations of the motions of Jupiter's satellites, as well 
 as of the satellites of the other planets, show that they con- 
 form to Kepler's third law when the planet is taken as the 
 centre of force, and hence that the forces which act on them 
 toward the planet are proportional to their masses. On the 
 assumption that the force with which the planet acts on the 
 sun follows the same law, the attraction between the sun and
 
 GRAVITATION Oli MASS ATTRACTION. 76 
 
 the planet is proportional to the mass of the planet and also 
 to the mass of the sun. 
 
 The arguments which have now been adduced lead to the 
 general conclusion that a force acts between any two portions 
 of matter in the universe, which tends to draw them together, 
 and that the magnitude of this force is proportional to each 
 of the masses and inversely proportional to the square of the 
 distance between them. This law is called the law of gravita- 
 tion or, from the fact that the force depends directly on the 
 masses of the interacting bodies, the law of mass attraction. 
 If we represent by m the mass of one of the bodies, by m', the 
 mass of the other, by r, the distance between them and by k, 
 a constant or factor of proportion, we may express the force 
 
 of gravitation F between these bodies by the formula F = k mm 
 
 The factor of proportion fc is the same in all cases. 
 ^" 53. Gravitation Constant. The factor of proportion which 
 Tias been introduced in the formula just given is called the 
 gravitation constant. It is manifestly equal to the force which 
 two unit masses will exert on each other when they are unit 
 distance apart. It may be determined by a direct determina- 
 tion of the gravitational force with which two bodies, whose 
 masses are known, act on each other. 
 
 An experiment designed for this purpose by Mich ell was 
 carried out by Cavendish in 1798. It is always known and 
 referred to as Cavendish's experiment. A horizontal rod, carry- 
 ing on its ends two equal balls of lead, was suspended at its 
 middle point from a long cylindrical wire. When not acted 
 on by any other forces than its own weight, such a system will 
 assume a definite position, in which there is no twist in the 
 wire. If the wire is twisted by turning the rod through any 
 angle in the horizontal plane, the twist in the wire will intro- 
 duce an elastic reaction proportional to the angular displace- 
 ment of the rod, and by preliminary experiments the magnitude 
 of the couple which will cause a given twist in the wire may 
 be determined. After such a determination had been made, 
 two large balls of lead were so placed on either side of the rod
 
 76 GRAVITATION OR MASS ATTRACTION. 
 
 as to attract the small balls in opposite directions, and so to 
 introduce a couple which twisted the wire. The magnitude of 
 this couple was measured for different distances between the 
 large and small balls, and thus the forces which they exerted 
 on each other were determined. Cavendish found that the 
 force between the balls varied inversely as the square of the 
 distance between their centres, and, by using balls of different 
 masses, that the force between them was proportional to the 
 masses. The values of the forces thus determined, introduced, 
 with the known masses of the balls and the distance between 
 their centres, into the formula expressing the law of gravita- 
 tion, lead to the determination of the gravitation constant. 
 The Cavendish experiment has been several times repeated, 
 with every precaution to insure accuracy. The results obtained 
 by different observers are fairly consistent with each other. 
 The value of the gravitation constant obtained from them, 
 when the unit of mass is the gramme and the unit of distance 
 the centimetre, is about sixty-five billionths of a dyne. 
 
 A method recently employed by von Jolly led to the same 
 result. In this method a sensitive balance was mounted high 
 in a tower and additional scale pans were suspended from it 
 by long wires. This balance was first used to show that the 
 weight of a body varies with its distance from the earth's 
 surface. A weight placed in one of the upper scale pans was 
 balanced by another weight in the other upper scale pan. This 
 second weight was then placed in the lower scale pan and 
 found to be heavier than before. Its increase in weight was 
 determined, and measured the increase in the attraction of 
 the earth upon it, due to its being brought nearer the earth's 
 centre. To determine the gravitation constant, a large block 
 of lead was placed under the weight in the lower scale pan, 
 which, by its attraction, caused a further increase in that 
 weight. This was carefully measured, and from it, taken 
 together with the known values of the attracting masses and 
 their dimensions, the gravitation constant was calculated. 
 
 54. Acceleration at the Earth's Surface Due to Gravity. 
 In studying the motions of bodies caused by the earth's at-
 
 GRAVITATION OR MASS ATTRACTION. 77 
 
 traction we may consider the earth as fixed, the movement of 
 the earth toward the common centre of mass, which occurs 
 when the body moves toward the earth, being so small that it 
 may be neglected. This being so, it follows from the law of 
 gravitation that all bodies near the earth's surface will have 
 the same acceleration. As has already been seen, this con- 
 clusion was reached by Galileo from observation. This ac- 
 celeration is represented by g. It is of great practical im- 
 portance to know its exact value, because it furnishes us the 
 most precise measure of force in absolute units. We can 
 determine the mass of a body very exactly, and use its weight 
 as a force. Then the force exerted by the body, expressed in 
 absolute units, is its mass multiplied by this acceleration. 
 
 Bodies fall so fast that it is extremely difficult to make 
 exact observations on them. A method has been devised for 
 this purpose, depending on the use of electric currents and 
 magnets, which yields fairly accurate values of g. Observa- 
 tions on bodies rolling down an inclined plane, or with the 
 Attwood's machine, will yield approximate values of this 
 constant. Its exact value is obtained universally by means of 
 the pendulum, first used for this purpose by Huygens. The 
 pendulum is used for this purpose in two forms. In one of 
 these, known as Borda's pendulum, the bob is a heavy sphere 
 and the suspension a long cylindrical wire. The wire is hung 
 from a knife-edge, which rests on agate planes. The effect of 
 the knife-edge on the period of the pendulum is avoided by a 
 preliminary adjustment. The efficient parts of the pendulum 
 are therefore regular bodies and the moment of inertia and 
 the static moment of the pendulum can be calculated. The 
 period of oscillation is observed, and, by the aid of the formula 
 given in 40, the value of g is determined. 
 
 In the other form, known as Kater's pendulum, the pendu- 
 lum is a stiff rod or bar carrying a heavy weight at one end 
 and furnished with two knife-edges which confront each other. 
 An additional movable weight is also carried on the bar. By 
 a suitable adjustment of this weight, the periods of the pendu- 
 lum swinging about the two knife-edges may be made the
 
 78 GRAVITATION OR MAS ATTRACTION. 
 
 same. The pendulum thus adjusted is a reversible pendulum, 
 and, as was seen in the discussion of 40, the distance between 
 the knife-edges is the length of the simple pendulum which 
 will swing in the same period. Having measured this distance 
 once for all, and having determined the period, the value of g 
 may be calculated by the aid of the formula for the simple 
 pendulum. 
 
 The value of g is different at different places, ranging be- 
 tween 978 at the equator and 983 at the pole, when the units 
 of length and time are the centimetre and second. For ordi- 
 nary calculations in our latitude we may use the value 980. 
 
 55. Density of the Earth. Newton did not attempt any 
 experimental demonstration of his hypothesis that an at- 
 traction exists between any two bodies. The experiment of 
 Cavendish by which this hypothesis was confirmed was made 
 long after his time. The attempts which were made to confirm 
 it, before Cavendish's experiment was executed, involved a 
 comparison of the attraction of the earth for a body with the 
 attraction exerted on that body by a portion of the earth, and 
 the numerical results obtained were worked over so that the 
 result stated was a value of the mean density of the earth. 
 
 The first of these experiments was made by the astronomer 
 Maskelyne in 1774. The theory of this experiment is as follows: 
 If two stations on the same meridian are chosen and the lati- 
 tudes of these stations observed, the difference of latitude will 
 plainly be equal to the angle between the radii of the earth 
 drawn from its centre to the two stations, if the plumb-lines 
 with reference to which the latitudes are measured are in the 
 directions of those radii. If, however, a great mass, like a 
 mountain, stands between the two stations, the plumb-bob 
 south of it will be drawn toward the north so that the latitude 
 measured at that station will be less than the true latitude. 
 The plumb-bob north of the mountain will be drawn toward 
 the south so that the latitude measured at the northern station 
 will be greater than the true latitude. If we know the true 
 difference of latitude between the two stations, from a direct 
 measurement of their distances apart and our knowledge of
 
 GKAV1TATION OR MASS ATTRACTION. 79 
 
 the size of the earth, we may determine, by the observations 
 here indicated, the amount by which the plumb-bob has been 
 drawn aside by the attraction of the mountain, and hence may 
 determine the relative values of the attraction of the mountain 
 and of that of the earth. We may then determine the actual 
 mass of the mountain from its dimensions and the average 
 density of its parts, obtained from a study of specimens of 
 the materials which compose it. From these data, assuming 
 the law of gravitation, we may calculate the mass of the earth 
 and hence its average density. By an experiment like the one 
 here outlined, Maskelyne obtained for the mean density of the 
 earth the value 4.7. 
 
 Another essentially similar method was employed by Car- 
 lini in 1824. He observed the oscillations of a pendulum -at 
 the bottom and at the top of a mountain. Assuming the law 
 of gravitation, he calculated what the acceleration should be 
 at the top of the mountain if its difference from that at the 
 bottom depended solely on the change of level, and compared 
 this calculated result with the acceleration observed at the 
 top of the mountain. He found that the observed acceleration 
 was greater than the calculated one. The difference he 
 ascribed to the attraction of the mountain. By proceeding 
 from this point as in Maskelyne's experiment, he obtained for 
 the density of the earth the value 4.8. 
 
 The experiments of Cavendish and of von Jolly may also 
 be used in the determination of the earth's density. The re- 
 sults obtained in these experiments are much more consistent 
 with each other that those obtained by methods in which the 
 attraction of large parts of the earth is observed, and may be 
 received with greater confidence. The value of the mean 
 density of the earth obtained from them is about 5.64.
 
 80 KLAST1CITY. 
 
 ELASTICITY. 
 
 5G. Cohesion. When we try to break apart a piece of metal 
 or wood, we encounter a very considerable resistance. This 
 resistance is a manifestation of forces which act between the 
 parts of the body. If we make the supposition that the mat- 
 ter of the body is so distributed as to occupy a considerable 
 part of its volume, it may be shown by analysis that these 
 forces cannot be due to the attraction of gravitation between 
 the parts, the forces arising from gravitation being so small 
 as to be negligible in comparison with them. The parts of the 
 bddy cohere with each other, and the forces between them are 
 called forces of cohesion. They may be considered attractions 
 between the parts of the body. 
 
 When we attempt to compress the body into a smaller vol- 
 ume, we again encounter a resistance, which may be ascribed 
 to a repulsion between the parts of the body. It is not cer- 
 tain that these repulsive forces really exist. The effects 
 ascribed to them may also be ascribed to the motions of the 
 parts of the body, due to the heat of the body. We shall, 
 however, adopt this hypothesis of repulsive forces provision- 
 ally. If we do so, we must consider the form and size of a 
 body, in any condition in which its parts are relatively at rest, 
 to be determined by a balance between the attractions and 
 repulsions in every part of it. 
 
 For a reason which will appear hereafter these forces be- 
 tween the parts of the body are often called molecular forces. 
 
 57. Elasticity. If a metallic w r ire is fastened at one end, 
 and a weight hung on the other, the wire will stretch a little. 
 The addition of another weight will stretch it further. When 
 the weights are removed, provided the wire has not stretched 
 beyond a certain limit, its length will become what it was at 
 first. Phenomena essentially similar are exhibited by a rod 
 when it is bent, a wire when it is twisted, a column on top of 
 which a heavy weight is placed, or a mass of water or air
 
 ELASTICITY. 81 
 
 confined in a vessel and subjected to pressure. In each case 
 of this sort, the forces which counteract those applied to the 
 body are evidently the forces of cohesion and of repulsion 
 which have just been considered. The system of forces or 
 pressures applied to the body may be called in general a stress. 
 The change in shape or size or in both which the body under- 
 goes may be called its deformation or strain. As a result of 
 common observation we may say that a stress applied to a 
 body always produces a strain in it. If the body recovers its 
 original condition, either in whole or in part, when the stress 
 is removed, it is called an elastic body. The forces which it 
 exerts against the applied stress are called elastic forces, and 
 the elasticity of the body is the property which it possesses of 
 exerting such forces. So far as our experiments go, there is 
 no body which does not possess elasticity, if not with respect 
 to all stresses, at least with respect to certain types of stress. 
 
 58. Hooke's Laiv. In 1679 Hooke gave an account of ex- 
 periments which he had tried on the stretching of wires by 
 weights. He found that for wires of the same length and 
 thickness, and of the same material, the elongations produced 
 by the addition of different weights were proportional to those 
 weights. He stated this result of his observations in the law, 
 Ut tensio, sic vis; the extension is proportional to the force. 
 This law is of far wider application than may be thought from 
 the way in which it was derived. It applies to all cases of 
 strains produced by stresses. If the stress is of a certain 
 type, it will produce a strain of the same type. Thus, for ex- 
 ample, a pull on a wire will lengthen it; a couple applied to 
 one end of a wire, whose other end is fixed, will twist it; a 
 pressure applied to a mass of water will compress it. In these 
 cases, and in all others which might be cited, the law holds 
 that the strain is proportional to the stress. Hooke's law is 
 therefore the general law of elasticity. 
 
 59. Compression and Expansion. A pressure applied to a 
 surface so as to be equal at every point on that surface, and 
 normal to it, is called a hydrostatic pressure. It may be either 
 positive or negative. The pressure directed outward from
 
 82 ELASTICITY. 
 
 the surface is taken as positive; that directed inward is nega- 
 tive. The bodies which we consider, in our present discussion 
 of elasticity, are supposed to be homogeneous and isotropic; 
 that is, they are alike in every part and possess similar prop- 
 erties, with respect to elastic forces, in every direction. A 
 hydrostatic pressure applied to such a body will change its 
 volume without changing its shape. When .the pressure is 
 positive, the volume of the body increases, or the body under- 
 goes expansion. When the pressure is negative, the body 
 undergoes compression. In the case of compressions and ex- 
 pansions, Hooke's law may be stated by saying that the 
 change of volume of the body is proportional to the change 
 in the pressure applied to it. 
 
 The change in volume of a body, caused by a pressure, is 
 manifestly proportional to the number of units of volume in 
 the body. Bodies of different sorts exhibit characteristic 
 changes of volume for the same pressure changes. "When the 
 pressure changes by one unit, the decrease or increase of a 
 unit volume will measure the compressibility or expansibility 
 of the substance composing the body. This quantity, while it 
 is the one most directly open to experimental examination, is 
 not the quantity commonly employed to represent the char- 
 acteristics of the body under pressure. We use instead the 
 ratio of the increase in pressure to the change of volume of a 
 unit volume. This quantity is called the volume elasticity, or 
 often simply, the elasticity, of the substance composing the 
 body. 
 
 60. Elasticity of Traction. When one end of a body is 
 fixed, and a force is applied to the other end to lengthen it or 
 to shorten it, the force applied is called a traction. The 
 tractions commonly observed are pulls which lengthen the 
 body, and the bodies used are long thin ones, like wires or rods. 
 For such a body Hooke's law takes the form that the elonga- 
 tion of the body, that is, its increase in length, is proportional 
 to the stretching force. When we examine wires of the same 
 material, but of different lengths and cross-sections, we find 
 that their elongations are proportional to their lengths, and
 
 ELASTICITY. 83 
 
 inversely proportional to their cross-sections. Wires of equal 
 lengths and cross-sections, but of different materials, will 
 exhibit different elongations when acted on by equal forces. 
 The preceding statements are all contained in the formula 
 
 e = _ , in which F represents the force, I, the length, 5, the 
 
 cross-section, and -,, ihe factor of proportion. This factor is 
 
 different for each substance, and is characteristic of it. It is 
 evidently equal to the elongation of a wire of unit length and 
 unit cross section stretched by a unit force. Its reciprocal n 
 is more commonly employed to characterize the elasticity of 
 a substance with respect to traction. This quantity is the 
 ratio of the force applied to the elongation which it will 
 produce in a wire of unit length and of unit cross-section. It 
 is called the modulus of fractional elasticity, or often Young's 
 modulus. It was introduced by Thomas Young in 1807 as a 
 convenient characteristic number for elastic bodies. 
 
 61. Elasticity of Torsion. When a wire clamped at one 
 end is twisted by a couple applied to the other end, a pointer 
 attached to it in the plane of the couple will turn through an 
 angle which is proportional to the moment of couple. The 
 wire is said to undergo torsion, and Hooke's law for the case 
 of torsion is given by the statement that the amount of 
 torsion is proportional to the moment of couple. Experiments 
 on wires of different lengths and cross-sections show that the 
 amount of torsion is proportional to the length of the wire and 
 inversely to the square of the cross-section. It also depends 
 on the material constituting the wire. The results of these 
 
 experiments are collected in the formula tf> = * ^ , in which 
 
 C represents the moment of couple, and _?T the factor of pro- 
 portion. The quantity n, called the modulus of rigidity, or 
 the rigidity, is different for different substances, and character- 
 istic of them.
 
 84 ELASTICITY. 
 
 02. Elasticity of Flexure. If a straight rod or bar is 
 clamped at one end, and if the other end is pulled aside by 
 any force, the bar is bent or flexed. The amount of the flex- 
 ure, determined by the distance through which the end of the 
 bar moves, is proportional to the force applied to produce it. 
 This is the statement of Hooke's law for this case. The 
 amount of flexure depends also on the dimensions of the bar 
 and on the material of which it is composed. The elastic co- 
 efficient or modulus characteristic of any particular substance 
 in this case is the same as the modulus of tractional elasticity. 
 That this should be the case may be seen if we consider a dia- 
 gram representing a bent rod. The line running down the 
 middle of the rod is" not changed in length by bending, while 
 those parallel to it above it are elongated and tho~se below it 
 shortened. As each of these lines reacts against the bending 
 force in a way that depends upon its elasticity of traction, it 
 is plain that their combined effect will be measured in terms 
 of that elasticity. 
 
 63. Types of Strains and Stresses. In most of the theo- 
 retical discussions of the properties of elastic bodies we limit 
 ourselves to the consideration of very small deformations. 
 We suppose these deformations to be such that a line in the 
 body which is straight before the deformation remains 
 straight after the deformation. 
 
 Analysis shows us that with these limitations any strain 
 may be obtained by the superposition of strains of two types. 
 These two types of strain, called compression and shear, are 
 distinctly different. 
 
 A compression is a change of volume, of such a sort that 
 the proportions of the parts of the body remain the same. 
 That is, it is a change of volume without a change of shape. 
 It may of course be either positive or negative. A shear is a 
 change of shape without a change of volume. This may be 
 brought about by a sliding of contiguous planes in the body 
 over each other, or by an elongation of parallel lines in the 
 body and a contraction of lines in some one direction perpendicu- 
 lar to them.
 
 ELASTICITY. 85 
 
 Corresponding to these two types of strain, there are two 
 types of stress, the hydrostatic pressure and the shearing 
 stress. The hydrostatic pressure has already been denned. 
 The shearing stress is a pair of equal and opposite pressures 
 applied tangentially to parallel surfaces in the body, combined 
 with another pair of pressures similarly applied to two other 
 surfaces in the body, at right angles to the first, in such a 
 way that the system of four pressures introduces no moment 
 of couple and serves merely to deform the body. Correspond- 
 ing to the two fundamental types of stress and the two funda- 
 mental types of strain which they produce, we have the two 
 fundamental moduli of elasticity, the elasticity of volume and 
 the rigidity. Other moduli of elasticity, in particular Young's 
 modulus, are functions of these fundamental moduli. 
 
 04. Failure of Hooke's Law. No mention has been made, 
 up to this point, of the fact that Hooke's law is not uni- 
 versally true. It is plainly not true for any strain whatever, 
 because, as we well know, a body to which sufficient stress is 
 applied may be permanently deformed or broken. When ex- 
 periments are tried with stresses of different magnitudes, it 
 is found that Hooke's law holds without perceptible error for 
 small stresses, but fails when the stress exceeds a certain 
 limit. The laws which have been stated in the foregoing sec- 
 tion must be interpreted in accordance with this statement. 
 
 The ideal body which obeys Hooke's law for all stresses, 
 and which returns precisely to its original condition when the 
 stress is removed, is called a perfectly elastic body. No real 
 bodies are perfectly elastic with respect to both of the funda- 
 mental types of strain. Liquids and gases, possibly also solids, 
 are perfectly elastic with respect to compression. No known 
 body is perfectly elastic with respect to shear. 
 
 The behavior of elastic solids under the action of forces, or 
 which have been subjected to forces for a time and then re- 
 lieved of them, depends in a curious and very complicated way 
 upon the time during which the force acts and upon the time 
 which elapses after the force is removed. The phenomena 
 exhibited by a body in these circumstances, and in others gen-
 
 86 ELASTICITY. 
 
 erally similar to them, are ascribed to what is called elastic 
 fatigue. 
 
 65. Solids and Fluids. Substances differ characteristically 
 in the way in which they behave under stress. In particular, 
 we find by experiment that all known bodies may be grouped 
 in one of two classes, according to the way in which they be- 
 have under shearing stress. The bodies in these two classes 
 are called solids and fluids. A solid is a body which will offer 
 a permanent resistance to a shearing stress; that is, it pos- 
 sesses a rigidity which depends upon permanent and conserva- 
 tive forces acting betM-een its parts. The fluid, on the other 
 hand, will not offer a permanent resistance to a shearing 
 stress. If the shearing stress, no matter how small it is, is 
 only applied long enough, the fluid will yield to it, will undergo 
 a continual and increasing deformation, and will not tend to 
 recover its original shape when the stress is removed. That 
 is, the fluid possesses no true rigidity. 
 
 It is not to be understood that every fluid will yield 
 instantly and completely to a shearing stress. All fluids, even 
 those whose parts move easiest among themselves, exhibit 
 viscosity or internal friction. In many fluids the viscosity ia 
 exceedingly great. When a shearing stress is applied to a 
 fluid, the rate at which the fluid yields to it depends upon its 
 viscosity, and the time taken to effect a perceptible deforma- 
 tion may be very great.
 
 CAPILLARITY 87 
 
 CAPILLARITY. 
 
 66. Capillarity. If we dip one end of a glass tube in water 
 and examine the water's surface around it and in it, we find 
 that it is not everywhere level, as our hydrostatic theory as- 
 serts that it should be. Around the walls of the tube, both 
 without and within, the water rises above the general level. 
 If the diameter of the tube is small, the whole free surface 
 within the tube rises above the general level, so that a column 
 of water is lifted by it. This phenomenon is said to have been 
 studied first by Leonardo da Vinci. The tubes in which it 
 appears conspicuously have a very fine, or capillary bore. The 
 general subject which deals with this phenomenon and with 
 many others essentially like it, and due to the same general 
 cause, is therefore named capillarity. 
 
 67. Surface Tension. The general laws of hydrostatics 
 depend upon the principle that a liquid subject to the attrac- 
 tion of gravity will be in equilibrium only when its configura- 
 tion is such that the action of gravity on it introduces no 
 shearing stresses. Now gravity is not the only force which 
 acts on the liquid. Its parts also exert forces of cohesion on 
 each other, and true equilibrium will not be reached until the 
 liquid assumes such a position that these forces of cohesion so 
 act, together with the weights of the parts of the liquid, as to 
 introduce no shearing stresses. We can explain all the 
 phenomena which are treated under capillarity by taking 
 these forces of cohesion into account. It is not necessary for 
 us to know, and in fact we do not know, the way in which the 
 cohesion depends upon the masses of the interacting parts and 
 the distances between them. It is necessary, however, to as- 
 sume this much, that the force of cohesion exerted by a small 
 part or element of the body only acts on those elements which 
 are in its immediate neighborhood. That is, we assume, as 
 the law of the force of cohesion between elements, that the 
 force between contiguous elements is very great, and dimin-
 
 88 CAPILLARITY. 
 
 ishes very rapidly when they are separated, so as to become 
 imperceptible, even when the distance between them is still 
 very small. This general law of cohesive forces is illustrated 
 by the behavior of an iron bar when we break it. It requires 
 a very great force to break it, but after it is broken, even 
 though the two surfaces at the break are fitted together again 
 with the utmost nicety, the two parts can be separated with 
 no perceptible effort. 
 
 Because of the short distance within which an element of 
 the liquid acts on its neighbors, those elements which lie 
 below the surface by a depth equal to this distance, which we 
 call the range of action, are in equilibrium under the action 
 of their neighboring elements. It is only those elements which 
 lie in or very near to the surface which are attracted unequally 
 in different directions. These unequal attractions, acting on 
 the elements of the surface film, will produce a peculiar con- 
 dition in it. This will be the same for all parts of the surface, 
 owing to the minuteness of the range of action, so long as the 
 radius of curvature of the surface is not very small, that is, 
 is not of the same order of magnitude as the range of action. 
 Thomas Young suggested that the special action of the co- 
 hesive forces in the surface film may be represented by sup- 
 posing the film to be under tension, similar in general to that 
 in a stretched membrane. This tension should be the same, in 
 the case of any given liquid, for all parts of its surface, it 
 is called the surface tension, and its numerical value, when 
 determined for the surface of separation between any two 
 bodies, is a characteristic number for those bodies. The 
 position of the liquid column in the capillary tube, or any of 
 the other phenomena ascribed to capillary action, are, on this 
 view, due to equilibrium between the weights of the parts of 
 the liquid and the forces due to the tensions acting in curved 
 portions of the surface film. 
 
 Another useful concept for the study of capillary phenom- 
 ena was introduced by Gauss. It is now called surface energy. 
 It is plain from the consideration of the action of the cohesive 
 forces, that when the surface of a liquid mass is enlarged, the
 
 CAPILLARITY. 89 
 
 potential energy of the liquid is increased on that account. 
 For, the surface can only be enlarged by the passage of ele- 
 ments of the liquid from the interior mass into the surface 
 film. Each of the elements in the film is drawn inward toward 
 the interior by a force of cohesion, and hence negative work is 
 done on all the elements which pass out from the interior into 
 the surface film. The negative work thus done is equivalent 
 to an increase in the potential energy of the liquid. The sur- 
 face, therefore, possesses an energy peculiar to itself, propor- 
 tional to the extent of surface and characteristic of the bodies, 
 separated by the surface. For a given surface its numerical 
 value is the same as that of the surface tension for the same 
 surface. 
 
 By the aid of this concept of surface energy all the phenom- 
 ena of capillarity may be explained as illustrations of the 
 general principle that the potential energy of a system of 
 bodies tends to become the least possible. 
 
 68. Plateau's Experiments. If a limited portion of liquid 
 is so situated that its equilibrium does not depend upon its 
 weight, the cohesive forces will act alone to determine the 
 form which the liquid will assume when in equilibrium. In 
 order to examine the behavior of a liquid in these circum- 
 stances, Plateau mixed a quantity of alcohol and water, ad- 
 justing the proportions of the two ingredients until the dens?- 
 ity of the mixture was the same as that of olive oil. When 
 masses of olive oil were introduced into this mixture, they 
 had no tendency to rise or sink, their weights therefore did 
 not effect their equilibrium, and the cohesive forces were free 
 to act alone. Each of the masses of oil assumed a spherical 
 form, provided they were entirely free in the supporting mix- 
 ture. This result is consistent with our general law of equi- 
 librium; for, equilibrium can exist in a mass only when the 
 pressure is the same in it everywhere, and in order that the 
 pressure should be the same, the curvature of the surface film 
 should be the same everywhere. Or, looking at the same mat- 
 ter from the point of view of surface energy, the form which
 
 90 CAPILLARITY. 
 
 the mass will assume will be that having the leasl potential 
 energy and therefore the least surface. 
 
 If the mass of oil is not free, but suspended on a wire frame 
 work, its surface will assume different forms. Whatever be 
 the form of these surfaces, they must be such that tensions in 
 them of equal value will give rise to equal pressure within the 
 liquid mass. Analysis has shown that this condition is 
 attained when the curvature of the surfaces is such that the 
 sum of the reciprocals of the principal radii of curvature is 
 the same at every point on them. 
 
 Minimal or ruled surfaces nr represented by films of 
 soapy water. The simplest surface of this kind, the sphere, is 
 illustrated by the ordinary soap bubble. The weight of the 
 film is so small a force, in comparison with the forces intro- 
 duced by the surface tension, that the figure due to the sur- 
 face tension is scarcely distorted at all by the weight of the 
 film. If a wire framework is dipped in the soap solution and 
 taken out again, films of water will usually adhere to it. 
 When a part of the frame work lies in a plane, the film 
 attached to it is also plane. If it is distorted, the film exhibits 
 the peculiarities of the minimal surface. When three films 
 meet, they always meet along a straight line, and make equal 
 angles with each other. Four films or more meet in a point. 
 The angles between the films which meet in this manner are 
 determined by the general condition that the surface of the 
 films is a minimum. 
 
 69. Rise of Liquids in Tubes. When a thoroughly clean 
 glass rod is dipped in water, the water rises around its sides 
 and wets it. If the rod, on the other hand, is dipped in mer- 
 cury, the mercury is depressed around it and does not wet it. 
 We ascribe this different behavior in these typical cases to the 
 different relative values of the cohesive forces of water and 
 of mercury and the different values of the forces with which 
 the glass attracts those substances. To describe the wetting 
 of the rod by the use of the idea of surface tension, we may 
 say that the tension in the surface separating the glass and 
 air is greater than the sum of the tensions in the surfaces
 
 CAPILLARITY. 91 
 
 separating water and air, and glass and water, so that the 
 edge of the liquid, where it meets the glass, is pulled upward 
 by the strongest tension. It is probably true that for very 
 clean glass the result of this action is to cover the whole of 
 the glass with a very thin film of water. 
 
 If a narrow tube of circular bore is immersed in a liquid 
 which wets it, the liquid will rise in the tube. The summit of 
 the column of liquid is terminated by a cup-shaped or almost 
 hemispherical surface, called the meniscus. The height of the 
 column, which is in equilibrium between its own weight and 
 the upward pull on it, due to the surface tension, is inversely 
 proportional to the radius of the tube. This law is known as 
 Jurin's law. It can be seen at once without formal demon- 
 stration that the theory of surface tension leads to this law. 
 For, the surface tension which lifts the column acts on every 
 point of the circumference of the upper surface, or along a 
 line which is proportional to the radius of the tube. The col- 
 umn which it lifts has a weight proportional to the height and 
 to the area of its cross section; that is, proportional to its 
 height and to the square of the radius of the tube. Since the 
 upward pull due to the surface tension is equal to the weight, 
 the height of the column is proportional to the surface tension 
 and to the radius and is inversely proportional to the area of 
 its cross section, or to the square of the radius. Therefore 
 finally the height of the column is inversely as the radius of 
 the tube. 
 
 If a small portion of liquid is contained in a long capillary 
 tube whose bore is not cylindrical, but conical, it will move 
 toward the small end of the tube, because it is the end of the 
 column whose radius is less upon which the surface tension 
 acts more advantageously. If the column of liquid is in a 
 cylindrical tube, it will move toward the end whose surface 
 tension is greater. We may diminish the surface tension of 
 one end of the column by raising its temperature, or by cov- 
 ering it with a film of another liquid of lower surface' tension. 
 
 70. Bubbles and Drops. A bubble is a mass of air sur- 
 rounded by a liquid. The shape of the liquid surface is de-
 
 92 CAPILLARITY. 
 
 termined by the condition that the pressure at every point, 
 due to the surrounding liquid, in addition to the pressure due 
 to the surface tension, shall be equal to the pressure of the 
 air in the bubble. For bubbles formed within films in the air, 
 substantially the same conditions of equilibrium hold, except 
 that the weight of the film is almost negligible, and equi- 
 librium exists when the external atmospheric pressure, in addi- 
 tion to the pressure due to the surface tension, is equal to the 
 pressure of the air within the bubble. The bubble of typical 
 form is one made by introducing air under a flat glass plate 
 held horizontally in the liquid. The plate keeps it from rising, 
 and it assumes a condition of equilibrium determined by the 
 balance of pressures already described. When the bubble is 
 very minute, its form is approximately spherical. As more 
 air is introduced it widens out and increases in height. Its 
 height depends upon the surface tension of the liquid, and 
 tends toward a limiting value, as the volume of the bubble 
 increases, of not more than a few millimetres in any known 
 case. Any horizontal cross-section of the bubble is circular. 
 Its under surface is fiat, when the bubble exceeds a certain 
 size, and over that flat surface the pressure of the air within 
 the bubble is equal to the pressure of the liquid column whose 
 height is the height of the bubble. For points on the side of 
 the bubble, at which the surface is curved, an additional 
 pressure exists due to the surface tension. 
 
 A drop is simply an inverted bubble, a mass of liquid rest- 
 ing on a support and surrounded by air. A drop of mercury 
 on a flat surface is a typical form. With evident modifications, 
 what has been said about the bubble applies equally well to 
 the drop. Drops of water, or of other liquids which wet solids, 
 are usually limited in respect of the area of the base on which 
 they stand either by impurities on the surface of the solid, 
 which prevent its being wetted, or by abrupt changes in the 
 shape of the solid. The theory of the equilibrium of such 
 drops is essentially the same as that already stated, but special 
 cases sometimes need careful study in order to perceive that
 
 CAPILLARITY. 
 
 they are not inconsistent with the general principles already 
 laid down. 
 
 71. Surface Viscosity. The parts of some liquids move 
 with much less freedom among themselves than those of others. 
 The forces with which they resist the force which moves them 
 are non-conservative forces and do not tend to restore the parts 
 which have been displaced to their original position. They 
 are ascribed to the mutual forces between the parts, coupled 
 with the mobility of those parts. The cause of this action is 
 called internal friction or viscosity. 
 
 In many liquids, probably in all of them, the viscosity In 
 the surface film is different from that in the general mass. Its 
 value is often very much greater than that in the mass. The 
 value of a liquid, as a means of illustrating the principles of 
 capillarity, often depends very much on the fact that it 
 possesses a considerable surface viscosity. For example, the 
 surface tension of water is greater than that of any other 
 ordinary liquid, and pure water would therefore seem to be 
 the best liquid that could be used, in experimenting on capil- 
 larity. On the other hand the surface viscosity of water is not 
 great, and the parts of a film of pure water will move so 
 freely under their own weight that the film very soon becomes 
 somewhere thin enough to break. The addition of soap to the 
 water, so as to make a strong soap solution, diminishes the 
 surface tension, but very considerably increases the surface 
 viscosity, so that a film of soap solution, whose parts do not 
 move freely, will remain unbroken for so long a time that its 
 form and peculiarities may be studied.
 
 GASES. 
 
 72. Mature of a Gas. The experiment of Torricelli with 
 the barometer showed that it was possible to obtain a region 
 in space that was to all appearance void or empty of matter. 
 This region above the mercury of the barometric column. was 
 called the Torricellian vacuum. 
 
 It occurred to von Guericke that it might be possible to 
 produce a similar vacuum by withdrawing air from a closed 
 vessel by means of a pump. After several unsuccessful at- 
 tempts, von Guericke at last succeeded in doing this, if not so 
 completely as in the Torricellian vacuum, at least sufficiently 
 well to enable him to study the behavior of bodies in a place 
 void of air/ He showed that if a partially collapsed bladder, 
 tied tightly at the neck, so that the air in it could not escape, 
 was placed in the vessel or receiver of his air pump, and the 
 air removed from around it, the air in it would swell out until 
 the bladder seemed full. Thus the air in the bladder appeared 
 to expand in all directions. Its behavior in this respect is in 
 marked contrast to that of a solid or of a liquid placed in the 
 receiver, neither of which will expand in all directions when 
 the air is removed. A body which exhibits this property of 
 expansion in all directions, or, as it is sufficient to say, of ex- 
 erting pressure in all directions on the walls of any closed 
 vessel containing it, is called a gas. 
 
 73. The Air as a Fluid. A limited portion of air has 
 weight. This fact was recognized by Galileo, who attempted 
 to prove it, without complete success, by first weighing a glass 
 vessel full of air, and weighing it again after some of the air 
 had been expelled by heating it. Aristotle failed in an attempt 
 to prove the same thing. He weighed a bladder full of air, and 
 then after forcing the air out of the bladder, weighed it again. 
 Of course he found no difference between the two weights, be- 
 cause the air which was outside the bladder in the second trial 
 took the place of, and weighed as much as the air which was
 
 OASKS. 95 
 
 within it in the first trial. The experiment cannot succeed 
 unless the walls of the vessel are rigid, and unless the air can 
 be removed from its interior. By employing an experiment of 
 this sort, von Guericke established the fact that air has weight. 
 
 This being so, and air being a fluid, the pressure relations 
 of which are the same as those of a liquid, it follows that, with 
 certain modifications, hydrostatic theorems apply to air as 
 well as to liquids. In particular Archimedes' principle appliss 
 to botli classes of bodies. A body weighs less in air than it 
 would weigh in vacuum, by an amount equal to the weight of 
 the air which it displaces. In very exact weighing, it is there- 
 fore necessary to take account of the volumes of the bodies 
 which are weighed, and of the weights used, in order to obtain 
 the weights of the bodies in vacuum. 
 
 In 1755 Black discovered the gas called carbon di-oxide. 
 Soon afterward hydrogen gas was discovered. This was soon 
 followed by the discovery of many other gases, and by the 
 recognition of the fact that the vapors formed by the evapo- 
 ration of volatile liquids were in most essential respects sim- 
 ilar to gases. Thus to the two classes of bodies which had 
 long been recognized, solids and liquids, there was added a 
 third class, gases. For our present purposes, we may define a 
 solid as a body which will retain its form and volume un- 
 changed, under the action of its own internal forces. A liquid 
 is a body which will retain its volume unchanged, under the 
 action of its own internal forces, but which yields to shearing 
 stress, so that it cannot retain a definite form, except when 
 placed in some receptacle. The walls of this receptacle exert 
 pressure on it of such a sort as to annul the shearing stresses 
 in it. A gas is a body whose internal forces do not constrain 
 it to assume any definite volume or form. It expands in all 
 directions, so that it cannot be confined as a liquid can, in a 
 vessel open at the top. When confined in a closed vessel, it 
 exerts pressure upon every part of the surface which en- 
 velops it. 
 
 74. Boyle's Law. The experiments of von Guericke with 
 the air pump were repeated and amplified by Robert Boyle.
 
 His attention was thus attracted tu the relation between the 
 volume of a limited portion of air and the pressure upon it, 
 and it occurred to him that it might be worth while to obtain 
 a set of values of the volumes and of the corresponding pres- 
 sures. He did this by isolating a small quantity of air in the 
 short limb of a U-shaped tube by means of a quantity of mer- 
 cury in the bend of the tube. When this mercury was so 
 adjusted that its ends stood at the same level in both limbs of 
 the tube, the pressure on the enclosed air was that of the 
 atmosphere. When more mercury was poured into the long 
 limb of the tube the surface in the short limb rose and the air 
 was compressed. The pressure on it was that of the atmo- 
 sphere, increased by the pressure of a mercury column whose 
 neight was the difference of level between the surfaces of the 
 mercury in the two limbs of the tube. By means of a number 
 of measurements of this sort Boyle showed that the volume 
 of the air in the tube varied inversely with the pressure upon 
 it. To illustrate this statement we shall consider the air con- 
 fined in the tube under atmospheric pressure. We may take 
 the volume of the air in this condition as unit volume, and 
 the pressure of one atmosphere as unit pressure. If mercury 
 is then poured in until the pressure becomes two atmospheres, 
 the volume of the air is reduced to one-half its original vol- 
 ume. If the pressure is made three atmospheres, the volume 
 of the air is one-third its original volume, and so on. For 
 sach of these cases the product of the volume and the corre- 
 sponding pressure is equal to 1. This product might have had 
 any numerical value we pleased to give it, depending upon our 
 choice of the volume taken as unit volume and the pressure 
 taken as unit pressure. But whatever that value was, the 
 experiments show that the product of the volume and the 
 corresponding pressure will always be the same. We may, 
 therefore, express Boyle's law in another and more convenient 
 form, by saying that the product of the volume of a given 
 mass of air and the pressure upon it is constant. 
 
 Boyle's law was subsequently shown to hold not only for 
 air, but also for all gases. If equal volumes of different gases
 
 are taken under equal pressures, the volumes of all of them 
 will change, when the pressures are changed, according to the 
 same law. The law holds not only for the case in which the 
 pressure is continually increased, but also for that in which 
 the pressure is decreased. Gases, therefore, differ from solids 
 and liquids in that they all possess the same volume elasticity. 
 
 75. Cray-Lussac's Law. Gases expand when heated. This 
 fact was known to Galileo, who used it in the construction of 
 the first thermometer. Owing to various causes, the law of 
 this expansion was not discovered until many years after Gali- 
 leo's time. In 1809 Gay-Lussac showed that all gases expand 
 at the same rate as their temperatures rise. That is, when the 
 temperatures of different gases are raised from that of the 
 melting point of ice to that of the boiling point of water, 
 their volumes all increase by the same proportionate amount. 
 Gases differ from solids and liquids in that they all possess the 
 same coefficient of expansion with rise of temperature. 
 
 This law was discovered, though not published, by the 
 chemist Charles, the inventor of the hydrogen balloon. It is 
 therefore sometimes known as Charles' law. 
 
 76. Law of Combining Volumes. Avogadro's Law. As the 
 result of his experiments on the chemical combination of 
 gases, Gay-Lussac showed that, in the case of complete chem- 
 ical combination, the volumes of the gases which combine and 
 the volume of the resultant product, if it also is a gas, are in 
 a simple numerical relation to one another. For example, two 
 volumes of hydrogen will combine completely with one volume 
 of oxygen, and as the result of that combination, two volumes 
 of water vapor are obtained. The volumes are of course meas- 
 iired at the same pressure and temperature. The law thus 
 stated is called the law of combining volumes. 
 
 An explanation of this law was given in 1811 by Avogadro, 
 and in 1811 by Ampere. 
 
 It had for some time been believed that bodies did not con- 
 sist of matter continuously distributed, but that the matter 
 of a body was collected in separate particles. Newton 
 described these particles in the following words: "It seems
 
 probable to me that God in the beginning formed matter in 
 solid, massy, hard, impenetrable, movable particles, of such 
 sizes and figures and with such other properties and in such 
 proportion to space as most conduced to the end for which 
 He formed them; and that these primitive particles, being 
 solids, are incomparably harder than any porous bodies com- 
 pounded of them, even so very hard as never to wear or break 
 in pieces; no ordinary power being able to divide what God 
 Himself made one in the first creation. ... It seems to 
 me, farther, that these particles have not only a vis inertiae, 
 accompanied with such passive laws of motion as naturally 
 result from that force, but also that they are moved by cer- 
 tain active principles." The particles of gases were supposed 
 to be of this sort. 
 
 Avogadro perceived that the law of combining volumes 
 could not be explained by assuming the particles of the gas 
 to be such particles as these. He therefore assumed that the 
 particles which compose the gas, and which give to it its char- 
 acteristic physical properties, are combinations of two or more 
 elementary particles. When these elementary particles are of 
 the same kind, the substance made up of them is a chemical 
 element. When they are of different kinds, the substance is 
 a compound. Chemical combination then involves a breaking 
 up of the groups of each of the elements, and a combination 
 of the particles composing these groups with each other to 
 form new groups. The elementary particles are called atoms, 
 the groups composed of them, molecules. These names were 
 first given by Ampere. 
 
 Avogadro announced the law that equal volumes of differ- 
 ent gases, under the same conditions of pressure and tempera- 
 / ture, contain equal numbers of molecules. With this law as 
 a foundation he was able to explain the law of combining 
 volumes. 
 
 No such law as this can be shown to hold for solids and 
 liquids. Gases, therefore, differ from solids and liquids in that 
 equal volumes of them, under similar conditions, contain equal 
 numbers of molecules.
 
 GASES. 99 
 
 77. Molecular Theory of Matter. This theory of the con- 
 stitution of bodies, which has been forced upon us by the 
 study of gases, affords a complete explanation of all the chem- 
 ical reactions between bodies. We therefore conclude that a 
 homogeneous body is composed of similar molecules, and that 
 these molecules are composed of atoms. In a few cases a mole- 
 cule contains only one atom, or the molecule and the atom are 
 identical. Broadly speaking, the science of chemistry is con- 
 cerned with the study of the various elementary atoms and 
 of their possible combinations. The science of physics deals 
 with the properties of bodies in so far as they depend upon 
 the peculiarities of their molecules. 
 
 In most physical operations, the molecules of the bodies 
 which are studied are unchanged by those operations. In 
 chemical operations, the molecules break up, and new mole- 
 cules of other sorts are formed. The elementary atoms are 
 not appreciably changed in any ordinary chemical operations, 
 and, until very recently, they have always been assumed to 
 be indestructible. But the study of radioactive bodies has 
 shown that it is highly probable that, at least in certain 
 cases, the atom is really a composite body, and that it may 
 change its character by the loss of some of its parts. We may, 
 for the present, assume the atom to be indestructible. 
 
 78. The Kinetic Theory of Gases. The properties possessed 
 in common by all gases indicate a common cause to which 
 those properties may be ascribed. As far back as the year 
 1738, an attempt was made by Daniel Bernoulli to explain 
 them as the result of the purely mechanical impact of the 
 particles or molecules of the gas. Bernoulli assumed that the 
 molecules of a gas are in constant motion, and that they col- 
 lide with each other and with the walls of the vessel which 
 contains them. He also assumed that their number in any 
 ordinary volume is enormously great. It is evident that the 
 effect of their collisions with each other will be to alter the 
 velocities of the individual molecules, so that, even if they 
 were equal at any time, they would not long remain the same. 
 For the purposes of elementary calculation, we may assume
 
 100 GASES. 
 
 that the effect which they produce by their impacts against 
 the walls of the vessel which contains them is the same as 
 that which they would produce if they all had a common veloc- 
 ity. The theory asserts that the pressure of the gas on the 
 walls of the containing vessel is due to the impacts of the 
 molecules. The problem then is to determine, on these sup- 
 positions, how the pressure on the walls w r ill depend upon the 
 volume of the vessel and the velocity of the molecules. Ber- 
 noulli solved this problem in the following way: 
 
 The gas is supposed to be contained in a cylindrical vessel 
 and confined within it by a piston, which at first stands at unit 
 distance from the base of the cylinder. If the piston is then 
 pushed down until its distance from the base is s, the volume 
 occupied by the gas may be represented by s, if its original 
 volume is represented by 1. It is assumed that the common 
 velocity of the molecules is not changed by this operation. 
 Now there are two reasons why the pressure of the gas should 
 be greater, when its volume is thus reduced, than it was be 
 fore: first, the molecules have been crowded together into a 
 smaller volume, and consequently there will be more mole- 
 cules lying near the walls; secondly, the average volume oc- 
 cupied by any one molecule is diminished, and consequently 
 the molecule will strike the side of this volume more fre- 
 quently. If we consider the first cause assigned, it appears 
 that the number of molecules which will lie contiguous to unit 
 area of the wall will be inversely proportional to the square 
 of the cube roots of the volumes of the gas, so that the ratio 
 of the pressures in the two conditions of the gas, so far is 
 
 they depend on the first cause, will be given hy s% : I. If we 
 consider the second cause assigned, it appears that the number 
 of impacts of any one molecule against the walls will be in- 
 versely as the average distance between the particles, or in- 
 versely as the cube roots of the volumes of the gas, so that 
 the ratio of the pressures in the two conditions of the gas, so 
 far as it depends on the second cause, will he sriven hy s^: 1. 
 When we consider that both the causes assigned act to-
 
 GASES. 101 
 
 gether, we conclude that the ratio of the pressures in the two 
 
 conditions of the gas is given by (s*) (s^):l, or by s:l, and 
 hence conclude, as the result of our original hypothesis as to 
 the nature of a gas, and the way in which it exerts pressure, 
 that when the velocity of the molecules of a gas is maintained 
 constant, its pressure varies inversely with its volume. 
 
 To show the way in which the pressure depends upon the 
 velocity of the molecules, when the volume is unchanged, we 
 notice that the impulse applied by each molecule to the wall, 
 measured by its change in velocity, is proportional to its 
 velocity, and that the number of impulses which each molecule 
 exerts upon the wall in unit time is also proportional to its 
 velocity, so that the effect due to any one molecule, and there- 
 fore the effect due to their combined action, is proportional to 
 the square of their velocity. If we assume that the velocity 
 of the molecules is increased by the introduction of heat, we 
 may explain in this way the increase of the pressure which a 
 gas exerts when its temperature rises, and by the use of Gay- 
 Lussac's law we may obtain a simple relation between the 
 velocity of the molecules and the temperature of the gas. 
 
 The success of the kinetic theory in explaining the behavior 
 of gases has been so complete, and its influence in establishing 
 the general kinetic theory of matter has been so great, that it 
 is worth while to discuss this question more formally, so as 
 to obtain the fundamental equation of the kinetic theory of 
 gases. We shall retain the same suppositions as those which 
 have been made already. Let us suppose that n molecules of 
 the gas are contained in unit volume, and that this volume is 
 a cube. The number of molecules contiguous to one face of 
 
 this unit cube will then be n%. The average volume occupied 
 
 bv each molecule is _, and the edge of the cube containing this 
 n 
 
 volume, or the average distance through which a molecule 
 moves in passing from one side of its molecular volume to the 
 
 other, is ( ~\ . A molecule which lies contiguous to the sur- 
 \/
 
 102 CASKS. 
 
 face will pass over t.wice this distance, with tlie average 
 velocity u. in the time which elapses between two successive 
 impacts. The time between these impacts is therefore given by 
 
 \ n ' The number of impacts which the tn< lecule will make 
 u 
 
 in unit time is the reciprocal of this, or _ 
 
 We now proceed to examine the effect produced on the wall 
 by the impact of a molecule. To do this, it is necessary to 
 make the supposition that the average energy of the molecule 
 is not changed by collision, so that the numerical value of its 
 velocity after collision with the wall is the same as it was 
 before collision. We consider a molecule moving directly 
 toward the wall. Its momentum before it meets the wall is 
 mu, and its momentum after it leaves the wall is of the same 
 numerical value and in the opposite direction. The total 
 change in momentum, which measures the impulse exerted by 
 the wall against the molecule, is therefore 2mu. 
 
 The molecules are moving in all directions, so that those 
 which lie contiguous to the wall are not all moving per- 
 pendicular to it at once. But we may assume that one-third 
 of them are moving in this way. The impulse applied to unit 
 surface, in the small time t, is equal to the impulse due to a 
 single molecule, multiplied by one-third of the number of 
 molecules which lie contiguous to the unit surface, multiplied 
 by the number of impacts which one molecule will make in 
 unit time, multiplied by the time t. This impulse is counter- 
 acted by and so is equal to the average force exerted by the 
 wall, multiplied by the same time t; and this average force 
 acting on unit surface measures the pressure exerted by the 
 wall, and therefore the pressure of the gas. Collecting the 
 
 f T 
 
 results of the previous paragraphs, we obtain pi = 2mu - '. !!!* . t, 
 
 3 *2 
 
 or p=$nmu*. The equation thus obtained is the fundamental 
 equation of the kinetic theory of gases.
 
 OA.SKS. 103 
 
 The number of molecules represented by n in this equation 
 is the number contained in unit volume. If we suppose the 
 amount of gas to remain unchanged, so that n represents the 
 number of its molecules in any circumstances, and suppose its 
 volume to be changed from unit volume to the volume v, the 
 
 number contained in unit volume will be ", and substituting 
 
 v 
 
 this for n in the fundamental equation for the pressure, we 
 obtain pv=$nmu*. The quantity on the right of this equation 
 remains a constant, if the common velocity of the molecules 
 remains constant. Hence we conclude that, on the hypotheses 
 assumed at the outset, the product of the pressure and volume 
 of the gas should remain constant, if the velocity of its mole- 
 cules is constant. This relation is that which was obtained 
 experimentally by Boyle. 
 
 We may extend this demonstration somewhat by making 
 an hypothesis about the velocities of the molecules, which more 
 nearly represents their true velocities than the one which we 
 have hitherto used. We suppose that the velocities are not 
 the same, but different. The number of molecules is, however, 
 so great that there will still be a large number. t whose 
 velocities lie very near a common value MJ. This set of mole- 
 cules will exert a partial pressure p t given by the formula 
 p^o = ^MjWietJ. Similar partial pressures will be exerted by other 
 sets of molecules, whose velocities lie near the values u 2 , u 3 , Ui, 
 
 The sum of all these partial pressures is the 
 
 pressure of the gas. If we represent by n the number of mole- 
 cules of the gas, by u a certain average velocity, by 15 n 2 , n s , 
 
 the numbers of molecules in the different sets 
 
 which have been considered, we may write an equation de- 
 finding the average velocity as follow? : mi 2 = n^ + w,rt* + n z u\ 
 
 + The average velocity thus defined is called 
 
 the velocity of mean square. In terms of this velocity we 
 obtain the equation pv=$nmn z for the pressure of the gas. 
 
 As has already been remarked, the equation just obtained 
 expresses Boyle's law. As will be seen later, when taken in 
 connection with Gav-Lussac's law, it affords us a measure of
 
 104 GASES. 
 
 temperature and an insight into the nature of heat. By the 
 aid of an additional theorem first given by Maxwell, it may be 
 made to demonstrate Avogadro's law also. Maxwell proved 
 that when two gases are mixed their most probable condition, 
 or their condition of final stability, is that in which the mean 
 kinetic energies of the molecules of both gases are equal. This 
 being so, if we consider equal volumes of two gases under the 
 same pressure and at the same temperature, so that they con- 
 form to the condition of being in mechanical and thermal 
 equilibrium with each other, the expression \nmu i is the same 
 for each. By Maxwell's law the factor iM a is also the same 
 for each, so that the factor n will be the same for each. Hence 
 Avogadro's law follows as a consequence of the kinetic theory. 
 It was shown by Joule that we may calculate the average 
 velocity of the molecules of a gas from the fundamental equa- 
 tion. Returning to the first formula, in which the volume of 
 gas is unity, the product nm of the mass of one molecule by 
 the number of molecules in unit volume is the density of the 
 gas. The pressure p corresponding to this density can be 
 measured, and hence the value of the average velocity calcu- 
 lated. Let us apply this equation to hydrogen under atmos- 
 pheric pressure. The pressure of one atmosphere is 1013373 
 dynes per square centimetre and the density of hydrogen under 
 this pressure and at the temperature of melting ice is 
 0.00008954 grammes per cubic centimetre. With these num- 
 bers we find that the average velocity of the hydrogen molecule 
 is 184260 centimetres per second. This is a little more than a 
 mile per second. From Maxwell's law of the equality of the 
 kinetic energies, the velocity of the molecules of any other gas 
 is to the velocity of the hydrogen molecule inversely as the 
 square root of the mass of the molecule of the gas is to the 
 square root of the mass of the hydrogen molecule. Thus if the 
 mass of the oxygen molecule is sixteen times that of the hydro- 
 gen molecule, the velocity of the oxygen molecule is one-quarter 
 that of the hydrogen molecule.
 
 10-1 
 
 FRICTION. 
 
 79. Resistance to Motion Due to Friction. When a block 
 of iron or wood is pushed over a table, the moving force is 
 always opposed by a resistance, or force in the direction oppo- 
 site to the motion. This resisting force may be called the 
 force of friction, or simply the friction between the two bodies 
 which slide over each other. The magnitude of the friction de- 
 pends upon the nature of the bodies and upon the state of 
 their surfaces, smooth bodies experiencing less friction than 
 rough ones. A solid of any form moved through a liquid, or 
 through a gas, encounters a similar resistance. In this case 
 the parts of the fluid which lie nearest to the solid are set in 
 motion by it, and as they in turn slide past other parts of the 
 fluid, they experience friction also, and communicate their 
 motion to those parts, so that the work which is done in 
 moving the solid against friction is largely spent in moving 
 the parts of the fluid. When the motion of the solid ceases, 
 the moving fluid is gradually brought to rest by the friction 
 between its parts. The friction between the parts of the fluid 
 is called internal friction or viscosity. The fact that bodies 
 moving through the air are resisted by the friction of the air 
 was known to Galileo, who used it to explain the fact that 
 very light bodies with large sxirfaces do not fall at the same 
 rate as heavy compact bodies do. This resistance of the air 
 must be avoided, or taken account of, in exact experiments on 
 the laws of falling bodies. 
 
 80. Friction Between Solids. The frictional resistance 
 which a solid body encounters when it is slid over a solid sur- 
 face is- believed to be due partly to the direct action of forces 
 between the molecules of the two bodies, and partly to the 
 partial interlocking of the small protuberances on tlie two 
 surfaces. The latter cause is more prominent when the sur- 
 faces of the bodies are rough, the former is, at least rela-
 
 106 FRICTION. 
 
 tively, more prominent when their surfaces are smooth or 
 polished. 
 
 The laws of sliding friction were investigated by Coulomb. 
 One of the two bodies which he employed was in the form of 
 a long horizontal plane, like a flat board. The other body 
 was pulled over it by a weight, and the observation consisted 
 in determining the value of the weight required to give the 
 moving body a constant velocity in different circumstances. 
 He found: 
 
 1. That the friction between two bodies is proportional to 
 the total pressure between the surfaces in contact. 
 
 2. That the friction is independent of the extent of surface 
 in contact, and so does not depend directly on the pressure 
 per unit area. 
 
 3. That the friction is independent of the velocity of the 
 moving bodies. 
 
 Friction is also exerted between two bodies which roll over 
 each other; that is, friction is exerted upon a wheel rolling 
 along a flat surface. Coulomb -found, for this case of rolling 
 friction, that the friction is proportional to the total pressure, 
 inversely proportional to the radius of the wheel, and inde- 
 pendent of its velocity. 
 
 In the case of sliding friction, the friction or the resistance 
 is measured by the weight required to maintain a constant 
 velocity in the moving body. The ratio of the total pressure 
 on that body to this resistance is called the coefficient of slid- 
 ing friction. The coefficient of rolling friction is given by an 
 essentially similar definition. 
 
 These laws of friction are only approximately true. It has 
 been found that the friction falls off very considerably when 
 the velocity of the moving body is very great. Its value prob- 
 ably changes also when the pressure is great. The coefficient 
 of friction is very much diminished by the use of lubricants. 
 
 81. Friction of Liquids. The internal friction of a liquid 
 was discussed theoretically by Newton, on the assumption 
 that it depends on the relative velocities with which contigu- 
 ous sheets of the liquid slide past each other. This sort of
 
 FRICTION. 107 
 
 motion is that which is produced by a shearing stress. Since 
 all liquids are viscous 3 that is, since they all offer more or 
 less resistance to this sort of motionj it cannot be said that 
 liquids offer no resistance to a shearing stress. The resistance 
 which they do offer, however, depends on the relative motion 
 of their parts, and since liquids do not possess true rigidity, 
 any shearing stress acting in them will produce motion, and 
 the liquid will gradually assume a form in which it is free 
 from shearing stress. 
 
 By the experimental and theoretical study of the effect of 
 friction on the motion of a solid in a liquid, or of the motion 
 of a liquid flowing pass a solid wall, or through a tube, it has 
 been found that the motion is affected as if the portions of 
 the liquid nearest the solid adhered to it without slipping, so 
 that the resistance to the motion is due entirely to the vis- 
 cosity of the liquid. When the floAV of a liquid through long 
 capillary tubes is treated on this assumption, it may be shown 
 that the amount of liquid which issues from the tube in unit 
 of time is proportional to the fourth power of the radius of 
 the tube. This theoretical conclusion is in complete accord 
 with the results of experiments carried out by Poisseuille. 
 
 The resistance to the motion of a solid through a liquid 
 increases with its velocity. When a force of a given value is 
 applied to a body to move it through a liquid, the velocity of 
 the body will gradually increase until it reaches a certain 
 limiting value, for which the resistance due to friction is equal 
 to the applied force. After that value has been reached, the 
 velocity will remain constant so long as the force is applied. 
 Thus a body falling through water will have a constant 
 velocity after it has fallen a certain distance. 
 
 82. Friction of Gases. In general what has been said about 
 the internal friction of liquids applies also to gases. The pe- 
 culiarity of the case of gases is this, that we can explain their 
 viscosity by means of the kinetic theory of gases. When one 
 layer or sheet of a gas slides past another, the molecules 
 which dart out from the more rapidly moving sheet into the 
 other one, carry with them a certain momentum in the direc-
 
 108 FRICTION. 
 
 tion in which that sheet is moving, and thus communicate 
 momentum to the more slowly moving sheet. The molecules 
 which dart out from the more slowly moving sheet into the 
 other one communicate to it a certain momentum in the 
 direction opposite to its motion. Thus the momentum of the 
 two sheets tends to become the same, or the sheets exert a 
 force on each other. Starting with this conception, the laws 
 of internal friction iu gases can be deduced, and shown to be 
 consistent with those obtained by experiment. In particular, 
 Maxwell, to whom the development of this theory is due, 
 showed that the internal friction of gases should be inde- 
 pendent of their densities. The experiments which Maxwell 
 and others carried out to test this conclusion showed that it 
 holds true within very wide limits. It fails only when the 
 density becomes exceedingly small. 
 
 The resistance offered by a gas to the motion of a body 
 through it depends on the velocity of the body. When the 
 velocity is small, like that of a swinging pendulum or a mag- 
 net, the results of experiment can be represented by supposing 
 the friction to be proportional to the velocity. For more 
 rapidly moving bodies, the friction seems to be nearly pro- 
 portional to the square of the velocity. For bodies moving 
 with the high velocities of the modern rifle ball or cannon ball, 
 the friction seems to be even greater than this law would in- 
 dicate. 
 
 The theory of the motion of a sphere through a gas leads 
 to the conclusion that the force required to overcome the re- 
 tarding effect of friction, and to keep the sphere moving with 
 a constant velocity, is proportional to the radius of the 
 sphere. If the sphere is falling through the gas, the force 
 which moves it, or its weight, is proportional to the cube of 
 the radius. It will fall with continually increasing velocity 
 until it reaches at last a limiting velocity, for which its weight 
 is equal to the resistance offered by friction. This limiting 
 velocity is therefore proportional to the square of its radius. 
 Thus the limiting or constant velocity attained by large spheres 
 falling through the air will be greater than that attained by
 
 FRICTION. 109 
 
 small spheres. Large drops of water which fall from a cloud 
 will have considerable velocity when they reach the ground, 
 while small drops will move toward the ground more slowly. 
 The very small drops which form a fog or cloud will fall so 
 slowly that their motion is hardly perceptible, and it can 
 easily be reversed or altered by the motion of currents of air.
 
 no 
 
 DIFFUSION. 
 
 83. Solution. If a lump of sugar is dropped into a vessel 
 full of water, it gradually disappears. The process by which 
 it disappears seems to be a gradual disintegration of those 
 parts of it which are nearest the water, and an absorption of 
 them into it. This process is called solution, and the sugar 
 is said to dissolve in the water. After solution the water has 
 acquired certain properties, which it did not possess before. 
 These are supposed to be due to the presence of molecules of 
 sugar distributed throughout the water. Very many bodies 
 will dissolve in water or in other liquids. The liquid in which 
 solution takes place is called the solvent, the body which dis- 
 solves in it, the solute. A limited amount of solvent will not 
 dissolve an unlimited amount of solute. After a certain 
 amount has been dissolved, the limit of solution has been 
 reached, and if more of the solute is present, it will remain 
 undissolved. The amount of solute which can be dissolved in 
 unit mass of the solvent is called its solubility. 
 
 In some cases there seems to be no limit to the amount of 
 solute which will be taken up by a solvent. Thus a homo- 
 geneous mixture may be formed by adding alcohol or glycerine 
 in any quantity to a limited quantity of water. In cases of 
 this sort, we speak of the one body as dissolved in the other 
 when only a relatively small amount of it is used, but in gen- 
 eral the two bodies are said to be mixed, and the body formed 
 is called a mixture. 
 
 We may explain the act of solution, in a general way, by 
 supposing that the molecular forces, acting between a mole- 
 cule of the solute and the neighboring molecules of the solvent, 
 are strong enough to tear away the molecule of the solute 
 from its original position and to transfer it tp the solvent. 
 It is uncertain whether the molecule of the solute, after it has 
 entered the solvent, is free, or whether it is permanently bound 
 to one or more molecules of the solvent, so as to form what
 
 DIFFUSION 111 
 
 is called a molecular aggregate. The weight of evidence is 
 at present in favor of the existence of such molecular aggre- 
 gates in many, if not in all, solutions. 
 
 84. Free Diffusion of Liquids. If a quantity of sulphuric 
 acid is placed in the bottom of a tall cylindrical jar, and if 
 water is then poured carefully into the jar, in such a way that 
 the sulphuric acid below is not disturbed, the water, being 
 relatively lighter, will float upon the acid, and the surface of 
 separation between the two liquids will be distinct. In pro- 
 cess of time, this distinct surface disappears, and a region 
 exists between the pure acid below and the pure water above 
 in which there is a mixture of water and acid. This region 
 gradually extends until at last it occupies the whole space 
 filled by the liquids. At first, the proportion of the two 
 liquids found in different parts of the column is very different, 
 but as time goes on, the mixture becomes more and more 
 homogeneous. In theory, however, it cannot become entirely 
 homogeneous until after an infinite time lias elapsed. This 
 process, by which one liquid mixes w.itli another, while the 
 mixture is not aided by currents set up in the mass, is called 
 diffusion, and a case of this particular sort, in which the two 
 liquids are not separated by any third body through which 
 they must pass to mix, illustrates free diffusion. 
 
 An experiment which is essentially similar may be tried 
 by placing pure water upon a solution of copper sulphate. The 
 dissolved salt will gradually rise through the pure water, until 
 the whole mass becomes a solution of copper sulphate of 
 uniform strength. 
 
 Any two liquids which may be mixed will diffuse into each 
 other in the way here described. When one of the two is a 
 solution, we consider the process to be the diffusion of the 
 solute from the parts of the mass in which its concentration is 
 greater to the parts in which its concentration is less. It was 
 assumed by Fick, and his assumption has been verified a? 
 approximately correct in many cases, that the rate at which 
 diffusion takes place along a line is proportional to the rate 
 of change of concentration along that line.
 
 112 DIFFUSION. 
 
 AVe may explain diffusion by ascribing it to a force exerted 
 upon the molecule of the solute by the molecules of the sur- 
 rounding solvent, it being supposed that those portions of the 
 solvent in which there are fewer molecules of the solute exert 
 the greater force. The motion which this force will set up is 
 resisted by the friction experienced by the molecule of the 
 solute as it passes through the solvent. Certain experiments 
 which have been made indicate that this friction is very great. 
 
 The process of diffusion may also be explained by supposing 
 it to be due, at least in part, to the motions of the molecules 
 of the solute. We have every reason to believe that these 
 molecules are constantly in motion, and that their motion is 
 in general similar to that already considered in our study of 
 gases. It is easy to see that the effect of such motions would 
 be to distribute the molecules of the solute throughout the 
 whole volume to which it has access. 
 
 85. Diffusion through Membranes. The earliest sys- 
 tematic study of diffusion was made on the diffusion of solu- 
 tions through solid substances. Many substances, as for ex- 
 ample, unglazed earthenware, and organic bodies, like parch- 
 ment paper or a bladder, or even india rubber in thin sheets, 
 will permit liquids to diffuse through them. If, for example, 
 a solution of common salt and water is placed in a bag made 
 of parchment, and immersed in a vessel of water, the salt will 
 gradually pass out through the bag, and water will enter it. 
 If another salt is dissolved in the water outside the bag, it 
 will generally enter the bag during the process. The rates 
 at which the two salts pass through the membrane depend upon 
 the nature of the salts and of the membrane. The general 
 process here described was called by Dutrochet osmosis. 
 
 In studying the phenomena of osmosis, Graham found that 
 bodies differed very remarkably with respect to the facility 
 with which they can pass through an organic membrane. He 
 divided all soluble bodies into two classes, to which he gave 
 the names crystalloid and colloid. Crystalloid bodies, like 
 sugar or salt, are those which in their solid state have gen- 
 erally a recognizable crystalline form. When dissolved in
 
 DIFFUSION. 113 
 
 water, they diffuse through aii organic membrane almost as 
 freely as if the membrane were not present. Colloid bodies, 
 like gum-arabic or glue, are amorphous, or have no crystalline 
 form. They scarcely diffuse at all through an organic mem- 
 brane. This distinction between crystalloids and colloids also 
 appears in the case of free diffusion, the colloids diffusing very 
 much slower than the crystalloids. Graham based on this 
 difference a process, called dialysis, of separating salts from 
 the colloid organic bodies with which they may be mixed. 
 
 86. Laws of Osmotic Pressure. In general, as has already 
 been described, osmosis takes place in both directions through 
 an organic membrane, and with such membranes no very 
 exact or general laws of osmosis can be discovered. Traube 
 discovered that membranes can be obtained, by taking advan- 
 tage of chemical action, which in many instances permit 
 osmosis in only one direction. Such membranes are called 
 semi-permeable membranes. If a small mass of sulphate of 
 copper is dropped into a dilute solution of ferrocyanide of 
 potassium, a film of cyanide of copper forms over its exterior. 
 Water can pass by osmosis 'through this film, but most sub- 
 stances which dissolve in water do not pass through it. A 
 film of this sort may be deposited inside the pores of an un- 
 glazed earthenware jar. If the jar is filled with a solution of 
 sugar, to take a specific example, and is partially immersed 
 in w r ater, the sugar will remain in the jar and water will enter 
 the jar from without. The membrane is permeable to water, 
 but is not permeable to sugar. 
 
 To study osmosis through semi-permeable membranes, a 
 small jar of the sort just described is closed tightly at the 
 mouth by a stopper, through which passes a long glass tube. 
 This arrangement may be called an osmotic cell. It was used 
 by Pfeffer in his researches. If such a cell is filled to the stop- 
 per with a solution of sugar, or of any ordinary salt, and is 
 immersed in water, the water will gradually enter the cell, so 
 that a column of liquid rises in the tube. This process may 
 continue for many days, but it ceases at last when the col- 
 umn has attained a height depending upon the concentration
 
 114 DIFFUSION. 
 
 of the solution. This height measures the pressure which will 
 force water out through the membrane as fast as it is coming 
 in by osmosis. The pressure thus measured is called the 
 osmotic pressure of the solution. If we arrange an osmotic 
 cell in such a way that pressure can be applied to the surface 
 of the solution in it, and so adjust this pressure, when the cell 
 is immersed in the water, that water neither enters nor leaves 
 the cell, the pressure thus determined is the osmotic pressure. 
 
 By the study of the results obtained by Pfeffer, van't Hoff 
 has shown that the magnitude of the osmotic pressure de- 
 pends upon the amount of solute in tne solution, in the same 
 way exactly as the pressure of a gas depends upon the amount 
 of gas enclosed in a given volume. In the first place, the 
 osmotic pressure of a solution of a given solute is proportional 
 to the concentration of the solution ; that is, to the amount of 
 solute in unit volume of the solution. If we consider the same 
 quantity of solute in solutions of different strengths, it is 
 plain that the coii-i-ntr<tti<>iis of those solutions an- invorsely 
 as their volumes. Thus the law just stated is equivalent to 
 the statement that when the quantity of solute is fixed, the 
 os-motic pressures of solutions of different strengths are in- 
 versely as their volumes. That is, the osmotic pressure obeys 
 a law like Boyle's law. 
 
 In the second place, for a solution of given strength, the 
 osmotic pressure rises with the temperature according to the 
 same law as that which determines the relation between the 
 pressure and temperature of a gas. That is, the osmotic pres- 
 sure obeys a law like Gay-Lussae's law. 
 
 In the third place, the osmotic pressures of many solutions 
 are equal when their molecular concentrations, measured by 
 the number of molecules of the solute contained in unit vol- 
 ume of the solution, are equal. That is, for these solutions, 
 the osmotic pressure obeys a law like Avogadro's law. In 
 many other cases, "however, this law ii? not fulfilled. Arrhenius 
 explained this fact by supposing that some of the molecules 
 of the solute, in these cases, are disintegrated into their con- 
 stituent parts. The number of independent portions of the
 
 DIFFUSION. 115 
 
 solute they cannot now be called molecules is thereby in- 
 creased, and the osmotic pressure increases likewise. The 
 molecules thus broken up are said to be dissociated or ionized. 
 This hypothesis not only accounts for the osmotic pressures 
 exhibited by such solutions, but also is consistent with and 
 explains many other of their peculiarities. 
 
 87. Diffusion of Gases. The free diffusion of gases goes on 
 in a way essentially similar to that of liquids. Its funda- 
 mental laws have been developed by Maxwell from the kinetic 
 theory. 
 
 The diffusion of gases through porous plates of earthen- 
 ware, gypsum, or graphite, was studied by Graham, who 
 showed that the rate of diffusion is proportional to the differ- 
 ence of the pressures on the two sides of the porous plate, and 
 is inversely proportional to the square roots of the relative 
 densities of the gases. Thus oxygen, which is sixteen times 
 as heavy as hydrogen, will diffuse only one-quarter as fast. 
 By taking advantage of this fact, we may partially separate 
 mixtures of different gases. These laws have been shown by 
 Reynolds to be consistent with the conclusions of the kinetic 
 theory. 
 
 Gases will diffuse through thin films of liquid, like the 
 walls of a soap bubble. They will dittuse also through certain 
 metals, when they are heated. Thus coal gas will diffuse 
 through the fire pot of a furnace, if it becomes red hot.
 
 116 
 
 SOUND. 
 
 88. General Considerations Respecting Sound. In all that 
 we have studied up to this point it has been evident that the 
 phenomena considered were directly presented by material 
 bodies. The actions of these bodies on one another are forces 
 which cause motions or bring about states of equilibrium. We 
 now are to consider another set of phenomena, associated with 
 the sense of hearing, which at first sight are not connected 
 with bodies at all. These phenomena are called sounds. 
 Although, when a sound is heard, there is nothing in the 
 sensation which we perceive which forces us to associate it 
 with the action of any external material bodies, yet universal 
 experience has taught us that sounds are associated with mat- 
 ter. When a sound is heard, we always believe it possible to 
 find some body which produces the sound, that is, which causes 
 the action to which our sensation is due; and we further be- 
 lieve that this action is transmitted to our ears from the body 
 which originates it by the action of intermediate bodies. From 
 the time of Aristotle it has been commonly believed that sound 
 is transmitted through the air as a series of disturbances or 
 impulses communicated to it by the body which is the source 
 of the sound. A rival theory, which assumed that sound is 
 transmitted by minute particles emitted from the sounding 
 body, had so little to support it that it was never given serious 
 consideration by physicists. All subsequent study has shown 
 that Aristotle's theory of the origin and the mode of trans- 
 mission of sound is correct, and the progress of knowledge has 
 resulted simply in more exact statements regarding the mode 
 in which the sounding bodies and the transmitting medium 
 act. 
 
 Confusion sometimes arises because the word sound is used 
 to designate a sensation received through our sense of hearing, 
 as well as the physical action by which that sensation is ex- 
 cited. In what follows we shall commonly use the word to
 
 SOUND. 117 
 
 denote the physical action, which can often be detected even 
 when it produces no audible sound. 
 
 Certain sounds produce in us a peculiar sensation which so 
 distinguishes them from others that they are called musical 
 sounds or tones. Such musical sounds excite a sensation 
 which lasts a perceptible time without perceptible change. In 
 many cases also the sensation produced possesses a certain 
 simplicity which distinguishes it from sounds of other sorts. 
 Other sounds, which do not possess these characteristics, are 
 called noises. We shall confine our attention almost altogether 
 to musical tones. 
 
 89. Sounds Produced ~by Strings. The use of stringed 
 musical instruments is of great antiquity. It must have been 
 observed, almost as soon as such strings were used, that the 
 tone produced by the string depends in some way on the force 
 with which the string is stretched and on the length of the 
 string. It is credibly reported that Pythagoras investigated 
 the relation between the length of a string and the pitch of 
 the tone emitted by it, and found that when strings whose 
 lengths were to each other as one, one-half, two-thirds, and 
 three-fourths, were under the same tension, the tones emitted 
 by them were those of the ordinary musical scale called the 
 fundamental, the octave, the fifth, and the fourth. This dis- 
 covery of Pythagoras is the first one recorded in the history 
 of physics, antedating by more than two hundred years any 
 recorded observations in mechanics. 
 
 Galileo and his friend Mersenne extended this observation 
 of Pythagoras by determining the lengths of string which pro- 
 duced all the tones of the ordinary musical scale. They also 
 recognized that the time taken by the string to execute one 
 vibration, called its period, or, what amounts to the same 
 thing, the number of vibrations executed by the string in one 
 second, which is the reciprocal of the period, depends on the 
 length of the string. They supposed that the period is directly 
 proportional to the length of the string, or that the number 
 of vibrations is inversely proportional to the length of the 
 string. We may therefore state the relation which they de-
 
 118 SOUND. 
 
 teimined between the tones of the musical scale and the 
 lengths of the strings which produce them as a relation be- 
 tween those tones and the number of vibrations producing 
 them. These relations are given for the major and minor 
 scales in the following tables: 
 
 MAJOR SCALE. 
 
 Name. do re rni fa sol la si do 
 
 Number of n - H 5n 4n Sn 5 15ra 2* 
 
 vibrations. 8 " 4 3 2 3 8 
 
 MINOR SCALK. 
 
 Name. la si do re mi fa sol la 
 
 Number of n _9 _^_ 4n 3n Sn_ ^n_ ^ 
 
 vibrations. 853256 
 
 On the assumption that the length of a string is inversely 
 proportional to its number of vibrations^ Mersenne deter- 
 mined the number of vibrations of a string emitting a partic- 
 ular tone, by comparing its length with that of a string so 
 long that its vibrations could be counted. 
 
 In all that has been said so far, the supposition has been 
 made that the strings are under the same tension. The pitch 
 of the tone emitted by a string depends, however, not only 
 on its length, but on its tension, on the material of which it 
 is composed, and on its thickness. Mersenne found, for strings 
 of the same material, that the number of vibrations is propor- 
 tional to the square root of the tension, is inversely propor- 
 tional to the length, and is inversely proportional to the 
 thickness. Later study has shown that if strings of different 
 materials are used, the number of vibrations which they pro- 
 d^uce is proportional to the square root of the tension, is 
 inversely proportional to the lengtn, and is inversely propor- 
 tional to the square root of the mass of the string in unit 
 length. 
 
 90. Transmission of Sound. After it had thus been shown 
 that a musical tone originates at a body which is executing 
 regular vibrations, attention was turned to the question of
 
 SOUND. 119 
 
 the transmission of these vibrations. They are manifestly 
 transmitted, in most cases, through the air. Other bodies, 
 however, may serve as the medium of transmission. Thus the 
 vibrations of a piano string may be transmitted to the audi- 
 tory nerve through a wooden rod, one end of which rests on 
 the lid of the piano, while the other end is held in the teeth. 
 The vibrations pass through the rod and through the bones of 
 the head, without passing through air. A scratch of a pin, 
 or a light blow of a hammer on one end of a long log, is 
 heard twice by a person at the other end. One of the sounds 
 is transmitted through the wood, the other is transmitted less 
 rapidly through the air. 
 
 The problem of the mode of transmission of sound was 
 first successfully studied by Newton. He assumed that the 
 sounding body emitting a musical tone is vibrating in such a 
 way that the motion of each of its parts has the essential 
 characteristics of the motion of a pendulum bob. That is, the 
 acceleration of each point on the vibrating body is propor- 
 tional to its displacement from the point in which it is at rest 
 in the undisturbed condition of the body. He assumed fur- 
 ther, that vibrations of a similar sort are transmitted to the 
 particles of air which are nearest to the body, from these to 
 the next set of particles, and so on. He showed that, on these 
 assumptions, the motion of any one particle of air will be a 
 vibratory one of the game period as that of the sounding body, 
 and that the motion transmitted from particle to particle will 
 proceed outward with a definite velocity. 
 
 We may describe the motion in the air by which sound is 
 transmitted, in the following way: Consider a long cylinder 
 or tube filled with air, and suppose that a layer of air at one 
 end of it is pushed forward. As it advances toward the next 
 layer of air, it exerts a pressure upon it, which sets that layer 
 in motion. The second layer in turn sets the third layer in 
 motion, the third layer the fourth, and so on; and thus a 
 forward motion is transmitted down the tube. If the suc- 
 cessive layers of air were rigidly bound together, the forward 
 motion of the first layer would be transmitted instantaneously
 
 120 SOUND. 
 
 through the whole column. In fact, the transmission of the 
 disturbance is not instantaneous, because the air is not incom- 
 pressible. The first layer in the column moves up toward the 
 second, the second up toward the third, and so on, producing 
 a region in which the air is more condensed than it is in its 
 condition of equilibrium. If the forward motion of the first 
 layer is checked after it has moved through a small distance, 
 this process of condensation ceases, and in turn the other 
 layers take up positions in which they are as far removed 
 from each other as they were before the disturbing motion 
 took place. The condensation originally produced proceeds 
 as a form of motion along the column. 
 
 If the first layer is riow suddenly drawn backward to its 
 first position, the second layer is pushed toward it by the pres- 
 sure on the other side of it, the third layer is pushed toward 
 the second, and so on. The movements of these layers like- 
 wise take place successively, so that while they are going on, 
 the air between the first layer and a part of the column which 
 the disturbance has not yet reached occupies a larger volume 
 than it does in its undisturbed condition, or is rarefied. When 
 the first layer has come to rest, the other layers successively 
 come to rest in the positions which they originally occupied, 
 and the rarefaction which has been produced proceeds up the 
 column. 
 
 Now, if the first layer of air is forced to execute regular 
 vibrations, like those of a pendulum bob, it will produce con- 
 densations and rarefactions similar to those which have been 
 described, which will succeed each other at regular intervals 
 and proceed up the column at a uniform rate as sound waves. 
 The time in which the exciting vibration is executed, or the 
 time between the production of two conditions of maximum 
 condensation, is called the period of the sound. The distance 
 between two successive points of maximum condensation is 
 the wave length. 
 
 When a sound is produced, as is commonly the case, in a 
 body of air extending in all directions, the successive con- 
 densations and rarefactions proceed outward from the body
 
 SOUND. 121 
 
 in all directions, and a particular condensation or rarefaction, 
 which has left the body at any instant, will be found, at a 
 later instant, on the surface of a sphere whose centre is in the 
 sounding body. The analogy between these spherical sound- 
 waves and the circular waves which are set up on the surface 
 of still water, by a slight disturbance made at a point in it, 
 was perceived by Yitnwius, although of course he had no 
 exact knowledge of the character of sound-waves. 
 
 If we fix on the most condensed portion of a particular 
 wave as a characteristic point in it, we may define the velocity 
 of a wave as the velocity of its most condensed portion. It ii 
 equal to the wave length divided by the period. This velocity 
 will plainly depend on the rate at which the successive layers 
 of air yield to the unequal pressure on the two sides of them 
 and this depends on the elasticity of the air and on the mase 
 of the particles which are to be moved, or on the density ot 
 the air. Newton showed that the velocity of a sound-wave 
 not only in air, but in any medium, is equal to the square root 
 of the elasticity divided by the square root of the density 
 In Newton's time, the elasticity of a medium which, like air. 
 obeys Boyle's law, was thought to be equal to its pressure 
 When Newton used this value of the elasticity to calculate the 
 velocity of sound, he obtained a value for it which was less 
 than that which had been obtained by direct experiment. This 
 discrepancy between theory and experiment was removed 
 many years later by Laplace, who called attention to the fact 
 that the elasticity of air is equal to its pressure only when 
 the temperature of the air is kept constant. Now it is known 
 that a sudden condensation of air will raise its temperature 
 and a sudden rarefaction will lower its temperature. The 
 condensations and rarefactions which constitute a sound-wave 
 pass through the air so rapidly that no time is given for the 
 equalization of the differences of temperature which they 
 produce, so that the elasticity which is to be used in the cal 
 culation of the velocity of sound is one determined on the 
 condition that heat is neither received nor emitted by the 
 air. The value of this elasticitv for air was measured and
 
 122 SOU ML> 
 
 found to be equal to its pressure, multiplied by the numerical 
 factor 1.41. With this value of the elasticity, the calculated 
 velocity of sound agreed with the experimental value. 
 
 To calculate the velocity of sound, we have given the 
 numerical factor 1.41, the pressure of one atmosphere, 1013373 
 dynes per square centimetre, and the density of air under the 
 pressure of one atmosphere and at the temperature of melting 
 ice, 0001^3 grammes per cubic centimetre. From these 
 data, we calculate that the velocity of sound equals 332.4 
 metres per second. The density of air equals its mass con- 
 tained in a given volume divided by that volume. If we sub- 
 stitute this value of the density in the formula for the veloc- 
 ity, we obtain the result that the velocity is equal to the 
 square root of the product of the pressure and the volume of 
 a given mass of air, multiplied by a factor which is a con- 
 stant. Now from Boyle's law we know that the product of 
 the pressure and the volume of a given mass of air is con- 
 stant. Therefore, the velocity of sound in air will be the 
 same, at the same temperature, whatever be its density. Thus 
 it will be the same at the highest altitudes as it is at the sea 
 level. From the same formula we learn that the velocity of 
 sound in different gases is inversely proportional to the square 
 roots of their densities. Thus in a light gas, like hydrogen 
 or coal gas, the velocity is greater than it is in air. 
 
 The velocity of sound was determined experimentally by 
 Mersenne, by noting the time which elapsed between the in- 
 stant at which the flash of a pistol was seen and the instant 
 at which the report was heard. By using sounds of different 
 intensities, produced by a cannon and a pistol, Gassendi proved 
 that the velocity of sound is independent of its loudness. That 
 it is also independent of its pitch is shown by the fact that 
 the different tones produced by the instruments of a band or 
 orchestra do not lose their harmonious relations when heard 
 by a distant observer. The statement that the velocity of 
 sound is independent of its loudness is not absolutely correct. 
 Captain Parry relates that when his men were engaged in gun 
 practice in the still air of the Arctic regions, the report of a
 
 * IUND. 123 
 
 gun was heard by a very distant observer before he heard the 
 word of command to fire. A profound study by Earnshaw of 
 the propagation of waves through air has shown that very loud 
 sounds should be propagated with greater velocities. 
 
 The mean of three of the best experimental determinations 
 of the velocity of sound at Cent, gives the value 331.5 
 metres per second. 
 
 91. Reflection of tiound-Waves. When a sound-wave in 
 air meets a solid obstacle, like a high wall, it is reflected from 
 it; that is, it is turned back in its course and proceeds from 
 then on in a reverse direction. From the analogous reflection 
 which may easily be observed when the waves on water en- 
 counter an obstacle, Vitruvius supposed such a reflection to 
 take place in the case of sound, and explained thereby the 
 echo. Observation shows that the sound-wave does not lose 
 its essential characteristics by reflection. 
 
 Reflection of sound occurs whenever the wave meets a sur- 
 face which separates the medium in which it is travelling 
 from another one. Two cases of reflection must be distin- 
 guished, which depend on the relations of the motions of the 
 two media when they are transmitting similar sounds. If 
 the second medium is made up of particles which move over 
 longer distances than those of the first medium do, when they 
 are transmitting similar sounds, and if we consider that por- 
 tion of the advancing wave in which the particles of tha 
 medium are being pushed forward, it is plain that when it 
 reaches the surface of separation, the last layer of particles of 
 the first medium will move farther out into the secpnd than 
 they would have done if the first medium had continued fur- 
 ther. With respect to the first medium, this movement of the 
 last layer may be looked on as composed of two movements, 
 one equal to that which the other particles of the first medium 
 have executed, and an additional movement in the same direc- 
 tion. This additional movement is of such a sort as to 
 start a movement in the first medium, proceeding in the op- 
 posite direction to that of the original wave. The successive 
 displacements caused by this movement are in the same direc-
 
 124 SOUND. 
 
 tion in space, or have the same sign, as the displacements in 
 the original movement by which they were produced. This 
 sort of reflection is therefore called reflection without change 
 of sign. 
 
 On the other hand, if the second medium is one whose 
 particles move less freely than those of the first medium, a 
 forward displacement is diminished when it comes to the 
 surface of separation. We may look on the forward displace- 
 ment of the particles of the first medium at that surface as 
 composed of two displacements, one equal to. the original for- 
 ward displacement, the other less than it in amount and in 
 the opposite direction. This displacement in the opposite 
 direction is of such a sort as to produce a movement in the 
 first medium, proceeding in a direction opposite tq that of the 
 original wave. In this case the reflection is said to be reflec- 
 tion with change of sign. 
 
 These two kinds of reflection are important in connection 
 with the study of the vibrations of sounding bodies. 
 
 92. Resonance. It was observed by Mersenne that if two 
 strings are tuned to the same pitch, and one of them is sounded, 
 the other will take up a vibration and will also emit a sound. 
 The experiment succeeds especially well if the two strings are 
 mounted on the same sounding board. It was also observed 
 that a string will be set in vibration if another string is 
 vibrating near it which is tuned to the octave or to the 
 twelfth of the tone which it will emit. This general phenom- 
 enon of the excitation of vibrations in bodies by the vibrations 
 of other bodies in their presence is called resonance. 
 
 The phenomenon of resonance is not exhibited by string* 
 only, but by all sorts of sounding bodies. Thus, one tuning 
 fork will respond to the vibrations of another which is in 
 unison or in simple harmony with it, and a column of air will 
 respond to the vibrations of a tuning fork or of a string. 
 
 To explain resonance we consider the effect of a succession 
 of small impulses applied to a string, which is capable of 
 regular vibration, at intervals which are equal to the natural 
 period of vibration. Such impulses are applied to it by the
 
 SOUND. 125 
 
 motions of the air, or of some vibrating body, like the sound- 
 ing board which supports it, and these are set up by the regu- 
 lar vibrations of the first sounding body. The effect of the 
 first impulse is a slight disturbance of the string. If that 
 impulse were the only one applied to it, it would execute a 
 number of minute vibrations in its own natural period. A 
 second impulse, however, is applied at the end of that period, 
 and therefore just at the time when the string is moving 
 through the position from which it was started by the first 
 impulse. The effect of the second impulse is added to that of 
 the first, and the vibration of the string is increased thereby. 
 As the successive impulses are applied to it, each at the time 
 when its action will increase the vibration, the vibrations be- 
 come greater and greater until they can often be perceived by 
 the eye and can be heard to give forth a musical tone. 
 
 If the natural period of the string is not equal to the in- 
 terval between the impulses, the effect of the first impulse will 
 again be to start a small vibration, but each impulse following 
 the first will be applied at a time which departs more and 
 more from that at which the string is in such a position that 
 its vibrations are increased by the impulses. For example, if 
 the string naturally executes 200 complete vibrations in a 
 second, and if the sounding body which sends impulses to it 
 executes 201 complete vibrations in a second, the second im- 
 pulse is applied one two-hundredth of a second before the 
 natural vibration of the string brings it to its most favorable 
 position, the third impulse is applied two two-hundredths of 
 a second too soon, and so on, the successive impulses being ap- 
 plied at times in which the position of the string is less and 
 less favorable. After 100 vibrations have been executed, the 
 impulse which tends to move the string in one sense is applied 
 to it just when it is moving in an opposite sense, and so its 
 effect is to destroy the original vibration. Thus the string 
 will not respond, and resonance will not occur, unless the 
 string has the same period of vibration as the sounding body. 
 
 When a string is made to respond to the vibrations of a 
 body which is emitting the octave of the fundamental tone
 
 J26 SOUND. 
 
 which the string emits, the tone emitted by the string is also 
 the octave. In this case, as was shown by Noble and Pigot, 
 and afterwards by Sauveur, the string is vibrating in two 
 portions. . Its middle point is at rest, and each half of it 
 vibrates as if it were an independent string, with its natural 
 period of vibration. When the resonance of a string is pro- 
 duced by a body sounding the twelfth of the fundamental 
 tone of the string, the responding string vibrates in three 
 equal parts. At two points, which divide it into these three 
 equal parts, the string is at rest. 
 
 These points at which the string is at rest, when it is 
 emitting a tone which is higher than its fundamental, are 
 called nodes. The vibrating portions between the nodes are 
 called ventral segments or loops. The tones which a string 
 will emit, when it vibrates in parts which are fractional parts 
 of its whole length, are called harmonics, partial tones, or 
 overtones. 
 
 An acute ear may detect that when a string is emitting its 
 fundamental tone, it is also emitting some of its overtones. 
 This observation was made by the early observers whose 
 names have already been mentioned. It shows that the vibra- 
 tion of a string may be much more complicated than the simple 
 vibration which we have assumed so far to be the one executed 
 by it. 
 
 In general, what has been said of a string with respect to 
 resonance and with respect to the production of overtones, not 
 only singly, but in combination with fundamental tones, may 
 be said also of all sorts of sounding bodies. 
 
 93. Sounding Bodies. The phenomena which have now 
 been described directed attention to the mode of vibration of 
 sounding bodies. It was at first assumed, as we have so far 
 done, that the vibration of each part of the body is a motion 
 similar to that of a pendulum bob, or, as we may call it, a 
 simple harmonic motion. The fundamental characteristic of 
 such a motion, w T hich takes place in a straight line, or in a 
 curve so slight that it may be considered a straight line, is 
 that the acceleration of the body is always proportional to
 
 SOUND. 127 
 
 its displacement from its position oi equilibrium, in which it 
 lies when the body is at rest. This kind of motion will be set 
 up in the body by its elastic forces, when the deformation of 
 the body is not great; for, as has already been seen in the 
 study of elasticity, Hooke's law holds, for all sorts of small 
 displacements, that the elastic force excited is proportional 
 to the displacement. 
 
 The simultaneous production of a fundamental tone and 
 of one or more overtones by the same body, forces us to 
 assume that the vibrations in it are not simple harmonic 
 motions, it was shown by Daniel Bernoulli that in this case 
 the vibration of a part of the body may be considered as aris- 
 ing from the coexistence in it of several simple harmonic 
 motions, whose periods correspond to those of the different 
 tones which the body is producing. The result of this com- 
 bination of simple harmonic motions is a periodic motion ; 
 that is, a motion which, however complicated it may be, re- 
 peats itself over and over again in successive equal periods. 
 In the case of a sounding body, this period is that of the 
 fundamental tone. 
 
 To prove this, Bernoulli demonstrated the principle of the 
 superposition of small motions. This principle may be de- 
 scribed as follows: If a part of an elastic body is set in 
 motion by two or more disturbances reaching it at the same 
 instant, its displacement will be the resultant of the displace- 
 ments which those disturbances would impart to it if they 
 were to affect it separately; and the disturbances which it 
 will cause in the parts of the body which lie near it are also 
 the resultants of those which it would have caused if it had 
 been affected by each of these disturbances separately. We 
 may illustrate this principle in certain special cases. For 
 example, if a sound wave is produced on one side of a room, 
 it will pass across the room and affect a portion of the air, 
 in a certain place and at a given instant, in a certain way. 
 The same portion of air will be affected also and differently 
 by another sound-wave starting from the other side of the 
 room. If both waves are set up so as to reach the chosen por-
 
 tion of air at the same instant, it will Le disturbed by both of 
 them, so that its resultant motion is a combination of the two 
 motions which they will separately produce. The two compo- 
 nents of this motion affect the air around this portion inde- 
 pendently, and each of the two waves travels on from the 
 point at which they cross each other without being changed 
 in any way. As another example, let us consider a point at 
 the middle of a vibrating string, at which two disturbances 
 from the opposite ends of the string arrive at the same 
 instant. These two disturbances may be such as to make the 
 resultant disturbance of the point double that which it would 
 acquire from either one of them, or they may be such, being 
 of the same magnitude and in opposite directions, that the 
 resultant disturbance at the point is zero, or the resultant 
 may have any intermediate value. In any case, after the dis- 
 turbances have passed the point, they will proceed unchanged 
 by any action of the one upon the other. 
 
 By the use of this principle of the superposition of small 
 motions, and of the laws of reflection, we may give a general 
 explanation of the vibrations of sounding bodies. The general 
 characteristic of such vibrations is that they take the form of 
 what are called standing or stationary waves in the body. 
 We may illustrate the formation of such waves by a special 
 example. Suppose that a stretched string is plucked near one 
 of its ends. The disturbance produced, which we may call an 
 elevation, runs along the string to the other end as a simple 
 wave, where it meets the rigid support in which that end of 
 the string is held. It is therefore reflected with change of 
 sign and comes back as a depression. It is again reflected at 
 the end from which it started, and becomes there an elevation, 
 which proceeds along the string again as has already been 
 described. The string is thus set in vibration. If the part of 
 the string which was first plucked is plucked again and again, 
 at times coinciding with those of the return of the reflected 
 wave to the place of origin, the successive disturbances will 
 be superposed and the vibration will be increased. If the 
 impulses are applied to the string twice as often, so that the
 
 SOUND. 129 
 
 advancing wave reaches the other end of the string just as 
 the second impulse is applied, the elevation produced by the 
 second impulse, and the depression due to the first impulse 
 after reflection, will meet in the middle of the string, and the 
 point at which they meet will not be disturbed. It will 
 remain undisturbed as the successive elevations and depres- 
 sions pass through it, while the parts of the string on either 
 side of it will vibrate. If the impulses are applie'd three times 
 as often as in the first case, there will be two points, dividing 
 the string into three equal parts, at which the advancing ele- 
 vations and returning depressions will meet, and which will 
 consequently remain at rest. Similar statements may evi- 
 dently be made for disturbances applied more frequently still, 
 if only the time between them is some fractional part of the 
 time required by the original disturbance to traverse the 
 length of the string. The points at which the string is at rest 
 are the nodes, the parts between them, the ventral segments. 
 The disturbance on the string by which its parts are kept in 
 uniform vibratory motion, is called the standing or stationary 
 wave. The disturbance usually impressed upon a string is not 
 an impulse at one point, but a general displacement of a con- 
 siderable portion of the string, or even of the whole string at 
 once. By the vise of the principle of superposition, we see 
 that the disturbance of each part of the string will be propa- 
 gated independently in the way already described for a simple 
 disturbance, and nodes will be produced when the period of the 
 original disturbance has any one of the values already indi- 
 cated in the preceding discussion. 
 
 In order to develop standing waves, it is not necessary to 
 apply to the string impulses which are properly timed. It is 
 sufficient to apply a large number of impulses, among which 
 there will be many which occur at the proper times. As has 
 already been explained, in 92, the string will respond to those 
 impulses, and will not be affected by the others. For example, 
 when a bow is drawn across a violin string, the impulses which 
 it applies to the string have no regularity, yet the string 
 responds to those which occur at intervals corresponding to its
 
 130 SOUND. 
 
 fundamental period, and also to those occurring at intervals 
 corresponding to the periods of some of its overtones. Those 
 overtones cannot occur which would establish nodes or fixed 
 points in that part of the string which is kept moving by the 
 bow. 
 
 The fundamental tone of a sounding body is determined by 
 the longest complete part of a standing wave, which can be 
 set up in it. The overtones which it will emit depend upon 
 the particular shorter standing waves which can be set up in 
 it. We may illustrate this general statement by the consider- 
 ation of certain special sounding bodies. 
 
 The vibrating string, which we have already described, is 
 fixed at both ends, so that reflection occurs at both ends with 
 change of sign. The time required by the string to execute a 
 complete vibration, or the period of the fundamental tone 
 which it will emit, is the time required by the disturbance set 
 up at one end of the string to pass from its point of origin to 
 the other end of the string and back again. That is, the period 
 of the fundamental tone is equal to twice the time required 
 for a wave to travel over the length of the string. The string 
 is half the length of the fundamental standing wave in it. Tt 
 may easily be seen that a node may occur in the middle of the 
 string, and that the vibrations of the halves of the string will 
 occur in half the fundamental period, so that the string 
 vibrating in this way gives the octave of the fundamental. 
 The string may also be divided by nodes into three parts, four 
 parts, or any number of fractional parts. To each of these 
 modes of subdivision there corresponds a tone which the string 
 can produce. Thus a stretched string will produce a funda- 
 mental tone and all possible overtones. 
 
 A column of air in a pipe may have standing waves set up 
 in it by a succession of properly timed impulses applied at 
 one end of the column, or if a number of differently timed im- 
 pulses are applied, as is clone in the case of the organ pipe by 
 blowing against the lip of the pipe, the column will respond 
 to certain of those impulses and standing waves will be set up 
 in it. If the pipe is open at both ends, the wave which ad-
 
 SOUND. 131 
 
 vances from the bottom of the pipe to the top, is reflected at 
 the top without change of sign. Thus an upward displacement 
 which leaves the bottom of the pipe is reflected as an upward 
 displacement. A standing wave will be established in the 
 column if the displacement which returns from the first re- 
 flection and is again reflected at the bottom as an upward dis- 
 placement is superposed on the next upward displacement 
 produced by the body which is setting up the disturbance. The 
 time between the two displacements is therefore the time 
 required for the wave to travel twice the length of the pipe. 
 The wave length in air of the fundamental tone which is 
 produced by un open pipe is therefore twice the length of the 
 pipe. The returning upward displacement, when it reaches 
 the middle of the pipe, meets there an advancing displacement 
 in the opposite sense, or a downward displacement, which has 
 left the lower end of the pipe at a time later by half a period 
 than that at which the original displacement left the same 
 place. These two displacements combine to produce no dis- 
 placement at the middle of the pipe, and the special result 
 here described holds generally for all the disturbances which 
 start from the lower end. They will always meet in the middle 
 of the pipe with reflected disturbances which are equal and 
 opposite to them. There will therefore be a node at the middle 
 of the pipe, when it is giving its fundamental tone. 
 
 If the impressed vibrations occur twice as often as those 
 which would give the fundamental tone, an upward displace- 
 ment will reach the upper end of the pipe just as the next 
 upward displacement is produced at the bottom. At the same 
 instant, a downward displacement will exist in the middle of 
 the pipe. This downward displacement moving up the pipe will 
 meet the reflected upward displacement at a point one-quarter 
 of the length of the pipe distant from the top. A node there- 
 fore exists at that point. The reflected upward displacement, 
 proceeding further down the pipe, will meet another advancing 
 downward displacement at a point one-quarter the length of 
 the pipe from the bottom. There is therefore a node at this 
 point also. The period of the tone emitted by this vibration
 
 132 
 
 is equal to the time taken by a wave to traverse the length 
 of the pipe. The wave length is equal to the length of the 
 pipe. The tone due to this vibration is the octave of the 
 fundamental. 
 
 Other vibrations will set up standing waves in such a pipe, 
 whose periods are one-third, one-fourth, etc., that of the funda- 
 mental. The general condition for the maintenance of stand- 
 ing waves in an open pipe is that there shall be the middle of 
 a ventral segment at each end, and that the distance between 
 a node and the middle of a ventral segment, or one-quarter 
 the length of the wave, shall divide into the length of the pipe 
 an even number of times. 
 
 If the pipe, instead of being open, is closed at the top, re- 
 flection will occur there with change of sign. At the open end, 
 where the disturbance originates, the reflection will still occur 
 without change of sign. At the closed end, the air is at rest, 
 and therefore that point in the column is a node. The open 
 end is at the middle of a ventral segment, so that the length 
 of the pipe is one-quarter the length of the wave which cor- 
 responds to its fundamental tone. Or more fully, if an up- 
 ward displacement leaves the open end, it traverses the pipe, 
 and is reflected at the closed end as a downward displacement. 
 A standing wave will be established when this downward dis- 
 placement, on reaching the bottom, is superposed on a down- 
 ward displacement caused by the body setting up the vibration. 
 This downward displacement will be set up at a time later by 
 half a period than the time at which the original upward 
 displacement was set up, so that the time taken by the dis- 
 turbance to traverse the pipe twice is equal to half a period. 
 
 If vibrations occurring twice as often are set up at the 
 open end, an upward displacement, which is reflected and 
 returns as a downward displacement, will meet another up- 
 ward displacement at the open end. These displacements, 
 instead of reinforcing each other, will destroy each other. A 
 closed pipe, therefore, will not give the octave of its funda- 
 mental. By reasoning generally similar to that already em- 
 ployed, it may be shown that the only overtones which the
 
 SOUND. 133 
 
 closed pipe can emit are those which correspond to waves 
 whose lengths are the uneven fractional parts of the funda- 
 mental wave length. The general condition for the mainte- 
 nance of standing waves in a closed pipe is that there shall be 
 a node at the closed end, the middle of a ventral segment at 
 the open end, and that the distance between a node and the 
 middle of a ventral segment shall divide into the length of 
 the pipe an uneven number of times. 
 
 A long rod of wood, glass, or metal, if clamped in the mid- 
 dle and set in vibration by stroking it along its length, will 
 execute vibrations and will maintain standing waves similar 
 to those of the open pipe. As in the case of the air in the pipe, 
 the length of the wave in the material of which the rod is 
 composed, whose period is that of the emitted tone, is twice 
 the length of the rod. An arrangement by which the velocity 
 of sound in a solid may be determined, depending upon this 
 fact, was invented by Kundt. It consists of a rod of the 
 material under investigation, clamped in the middle, and with 
 one end inserted lightly through a cork, which closes one end 
 of a glass tube. "Fine light powder is scattered within the 
 tube, and its other end is closed with a loosely fitting cork, 
 that can be moved up and down in it. When the rod is made 
 to vibrate longitudinally, vibrations of the same period are 
 impressed vipon the column of air in the tube, and by setting 
 the movable cork at the right place, they may be made to set 
 up standing waves in the tube. These waves stir up the 
 powder which has been scattered in the tube, and arrange it 
 in a regular pattern, from which the positions of the nodes 
 may be determined, and so the length of the wave in air, 
 whose period is that of the tone emitted by the vibrating rod. 
 Since the length of the corresponding wave in the rod is twice 
 the length of the rod, the ratio between twice the length of 
 the rod and the length of the wave in air is the ratio of the 
 velocity of sound in the rod to the velocity of sound in air. 
 If some gas other than air is used in the tube, the wave length 
 determined in it, compared with the wave length produced in 
 air by similar vibrations of the rod, affords a means of deter-
 
 134 
 
 mining the velocity of sound in the gas and so of testing New- 
 ton's formula. 
 
 If a rod is clamped at one end, it may be made to execute 
 transverse vibrations, which will, if sufficiently rapid, give 
 rise to a musical tone. When the rod is emitting its funda- 
 mental tone, {here is a node at the end at which it is fixed. 
 The first harmonic which is developed in such a rod is about 
 three octaves higher than the fundamental. 
 
 When a rod which is free at both ends vibrates trans- 
 versely, it develops two nodes, which are distant from the ends 
 about two-ninths the length of the rod. The rod may be sup- 
 ported at the points where these nodes occur without inter- 
 fering with its vibrations. If the rod is bent in the middle, 
 the two nodes approach each other, until, when the two 
 halves are parallel, the nodes are very near together. The 
 tuning-fork is an example of a rod bent in this manner. As 
 the ends of the fork swing toward each other, the middle of 
 the fork, to which is attached the stem or handle, moves down- 
 ward. As the ends move apart, the middle of the fork moves 
 upward. Thus, if the fork is sounding, and the handle is 
 brought down on the table, it will tap the table top at regular 
 intervals. The vibrations thus imparted to the table are com- 
 municated from it to the air, and thus the volume of sound 
 emitted by the fork is considerably increased. 
 
 Standing waves may also be developed in plates of metal or 
 glass, which are firmly clamped at some one point and are 
 bowed or stroked at some point on the edge. The nodes of the 
 waves thus prodxiced are detected by sprinkling a little sand 
 over the plate, which is thrown away from the vibrating parts 
 of the plate and collects at the nodes. This method of obser- 
 vation was introduced by Chladni, and the figures obtained 
 are known as Chladni's figures. 
 
 94. Quality of Sounds. When two musical instruments 
 are sounding the same tone, the sounds which they emit, how- 
 ever exactly they may agree in pitch, differ entirely in quality. 
 We may trace this difference in quality to the other tones be- 
 side the fundamental which the bodies are producing. To take
 
 SOUND. 135 
 
 the simplest case, tones of the same pitch coming from an open 
 and a closed organ pipe are perceptibly different in quality, 
 the tone from the open pipe being fuller and richer than the 
 other. This difference is due to the circumstance already dis- 
 cussed, that the column of air in the open pipe maintains 
 standing waves corresponding to all the overtones of the 
 fundamental, while that in the closed pipe maintains only 
 those corresponding to the uneven overtones. When a pipe is 
 sounded these overtones are emitted as well as the funda- 
 mental, and the quality of the tone is determined by their 
 presence or absence in the sound and by their relative in- 
 tensities. In the case of strings all overtones may be present, 
 and further additional tones occur, which are produced by 
 very rapid vibrations which differ considerably with the 
 material of which the string is made. It is in general true of 
 all wind instruments also that the tones emitted by them are 
 distinguished as respects their quality not only by the relative 
 intensity of their various overtones, but also by characteristic 
 tones, or even by characteristic noises, which depend upon the 
 material and the construction of the tube in which the air 
 column is contained, and also upon the way in which the 
 original vibrations are produced. 
 
 As a means of observing the presence of an overtone we 
 use an instrument designed by Koenig, called the manometric 
 capsule. This consists of a small box divided into two cham- 
 bers by a thin flexible membrane. One of these chambers is 
 kept filled with illuminating gas, which is burned in a small 
 jet at the end of a tube projecting from the chamber. The 
 other chamber is connected by a tube to the source of the 
 sound. When the sound is given forth, the gas jet is viewed 
 in a revolving mirror. The vibrations of the membrane due 
 to the sound produce changes in the height of the flame, and 
 the band of light which is seen in the mirror appears serrated 
 at the top. If the tube of the manometric capsule is intro- 
 duced into the side of an organ pipe, at a place where a node 
 exists, the alternations of rarefaction and condensation which 
 occur at an node will be demonstrated by a marked serration
 
 136 SOUND. 
 
 of the band of light. If, on the other hand, the tube is intro- 
 duced at the middle of a ventral segment, where there are no 
 changes of density, the serrations of the band will not appear. 
 
 The overtones of pipes may be produced with but little 
 admixture of the lower tones by changing the intensity of the 
 blowing and often by a manipulation of the mouth piece. The 
 existence of the nodes and ventral segments corresponding to 
 the overtones may be demonstrated by the manometric capsule. 
 
 By touching a string at the middle and bowing it, it can 
 be made to emit the octave of its fundamental. When it is 
 sounding the octave, a light rider of paper may be placed at 
 the middle point and will remain there undisturbed, whereas 
 a rider anywhere else on the string will be thrown off. Higher 
 overtones may be produced in a similar way by touching the 
 string at other points, and the existence of the nodes and 
 ventral segments corresponding to them may also be demon- 
 strated by the use of riders. 
 
 The most complete way by which to study a composite tone 
 is by the use of resonators. The resonator is a hollow metallic 
 sphere with a circular opening on one side, and on the other 
 a short projecting tube, to which the ear may be placed, or on 
 which the tube of the manometric capsule may be attached. 
 The air within the sphere, considered as a sounding body, has 
 a fundamental mode of vibration and but very few and feeble 
 harmonic vibrations. If this resonator is in the presence of a 
 body which is emitting the fundamental tone of the resonator, 
 the air in it will be set in vibration, and its vibration will be 
 indicated by the manometric capsule. A series of such 
 resonators may be made, tuned to the different tones of the 
 scale, or to the successive overtones of a fundamental tone, and 
 can then be used to investigate the different tones emitted at 
 the same time by a sounding body. It was by the use of such an 
 instrument, called an analyser, that Helmholtz demonstrated 
 the relation between the quality of a tone and the overtones 
 present in it. 
 
 95. Beats and Resultant Tones. When two tuning forks, 
 or two pipes, which have nearly the same pitch, are sounded
 
 SOUND. 137 
 
 together, the sound heard varies in intensity periodically. The 
 complete change of intensity which occurs, from the greatest 
 intensity heard through the least to the greatest again, is 
 called a beat. Beats may be explained in the following way: 
 The two sounding bodies are vibrating in periods which are 
 nearly, but not quite the same. At a certain instant they are 
 moving in such a way as to affect the air around them in a 
 similar manner. The disturbance in the air is then twice as 
 great as that which either of them would produce if it were 
 sounding alone. As they continue to vibrate, their motions 
 become less and less similar, until, after a certain time has 
 elapsed, they are exactly dissimilar and are affecting the air 
 around them oppositely. The sound then has its least in- 
 tensity. From that time on, the sound will gradually increase 
 in intensity, until one of the bodies has gained a whole vibra- 
 tion on the other, when the sound will again have its greatest 
 intensity. Since the time between two instants of greatest 
 intensity is that required for one body to execute one vibra- 
 tion more than the other, the number of beats heard in one 
 second will be equal to the difference between the vibration 
 numbers of the two bodies. 
 
 Koenig has shown that beats may also be heard when two 
 bodies are sounding together whose pitch is very different, 
 provided that the two tones are nearly an octave apart, or 
 are nearly in some other harmonious relation. 
 
 When two tones are sounded together whose vibration 
 numbers differ considerably, a third tone is often heard, called 
 the resultant tone. The vibration number of this tone is 
 equal to the difference between the vibration numbers of the 
 other tones. Koenig's observations make it probable that 
 resultant tones are caused by the regular beats that occur 
 when the two primary tones are sounded. Other resultant 
 tones are heard when two primary tones are sounded strongly, 
 which Helmholtz explained on the hypothesis that the move- 
 ments of the air set up by the sounding bodies are so great 
 that the simple law of elasticity is not longer applicable.
 
 138 SOUND. 
 
 90. Harmony. It is interesting to observe, that those 
 tones whose combination is most pleasing and gives most fully 
 the sense of harmony, are those whose vibration numbers are 
 in the simplest proportion to each other. Thus, next to the 
 combination of two tones which are in unison, or whose vibra- 
 tion numbers are the same, the most perfect harmony is 
 obtained when the octave is sounded with the fundamental; 
 that is, by the combination of two tones w T hose vibration num- 
 bers are as 2 to 1. The next most harmonious co'mbmation is 
 that of the fifth with the fundamental, whose vibration num- 
 bers are as 3 to 2. The combination of the fourth with the 
 fundamental, whose vibration numbers are as 4 to 3, and of 
 the third with the fundamental, whose vibration numbers are 
 as 5 to 4, are also harmonious. On the other hand, the com- 
 bination of the seventh with the octave, whose vibration num- 
 bers are as 15 to 16, is not harmonious. In this case beats are 
 heard. It seems probable that discordant combinations or 
 discords occur when beats are produced which lie between 
 certain limits. Tones which produce fewer beats than 10 per 
 second, or more beats than 70 per second, are not discordant, 
 but if the number of beats produced lies between those limits, 
 the tones are discordant. 
 
 97. Absolute S umber of Vibrations. So far we have con- 
 sidered mainly the relative number of vibrations executed by 
 a sounding body, or producing a sound, in terms of the vibra- 
 tions of some body taken as a standard. It is, however, a 
 matter of interest to determine the absolute number of vibra- 
 tions which corresponds to a certain pitch. The determination 
 of this number may be made in several ways. 
 
 It has already been explained how Mersenne, by using a 
 string so long that its vibrations could be counted, and com- 
 paring its length with that of a shorter string under the same 
 tension and emitting a standard tone, was able to determine 
 the number of vibrations corresponding to that tone. 
 
 Sauveur used the beats produced by two organ pipes, whose 
 vibration numbers were in a known ratio, to determine their 
 absolute vibration numbers. The pipes which he used gave the
 
 SOUND. 139 
 
 fundamental and the seventh of the next lower octave. Their 
 vibration numbers were therefore in the ratio of 16 to 15. In 
 the sound which was heard when they were sounded together 
 there were six beats per second, so that the difference of their 
 vibration numbers was 6. From these two relations it follows 
 that the vibration numbers of the two tones were 90 and 96. 
 
 Savart used a toothed wheel, which was rotated by a 
 mechanism, so constructed that its rate of rotation could be 
 controlled and that the number of rotations per second could 
 be counted. When the wheel was in rotation and a flexible 
 card brought up against the teeth, a musical tone was emitted, 
 and by regulation of the rate of rotation this tone could be 
 brought in harmony with that of a standard pipe or fork. The 
 number, of vibrations in the standard tone was then deter- 
 mined from the number of rotations made by the wheel in a 
 second, and the number of teeth on the wheel. 
 
 Duhamel set a light pointer on the end of a tuning fork, 
 and arranged it so that the fork when sounding was carried 
 along over a plate of glass covered with a thin coating of lamp 
 black. The pointer traced out a sinuous line on the blackened 
 surface, and the number of vibrations executed by the fork 
 was equal to the number of sinuosities in this line. By means 
 of an additional mechanism, a time-keeper was made to record 
 equal intervals of time on the same blackened surface, and by 
 counting the number of sinuosities lying between two marks 
 made by the time-keeper, the vibration number of the fork was 
 determined. 
 
 The vibration number of a tone may also be determined by 
 means of the siren. The siren is an instrument which pro- 
 duces a sound by means of the alternate emission and suppres- 
 sion of puffs of air at regular intervals. It consists of a box 
 or air chest with a flat top in which are pierced a number of 
 holes set uniformly .around a circle. A flat disk is mounted 
 on an axle, which turns in a support set at the centre of this . 
 circle, and the disk is set as near the top of the box as it can 
 be without touching it. A set of holes is made in the d 4 '*, 
 which correspond in number and position to those in the iid
 
 140 SOUND. 
 
 of the box. Sometimes, by setting the holes obliquely, the air 
 which is forced out of the box may be made to turn the disk, 
 but it is best lo turn the disk by some outside mechanism. A 
 counter is provided by means of which the rate of rotation of 
 the disk may be determined. When the disk is turned and air 
 is forced into the box, a puff of air "comes out of each hole 
 when the holes in the disk stand directly over those in the 
 box. When the disk turns on so that the holes in the box are 
 covered, the air is shut off. The number of puffs emitted dur- 
 ing one rotation of the disk is equal to the number of holes 
 in the disk. To determine the vibration number of a given 
 tone, the disk is turned until the tone emitted by the instru- 
 ment is in unison with the one whose vibration number is 
 desired. When this condition is obtained, the rotation, of the 
 disk is maintained uniform, and the number of its rotations 
 per second is determined by the counter. From this number 
 and the number of holes in the disk, the vibration number of 
 the tone may be obtained.
 
 141 
 
 HEAT. 
 
 98. Sensation of Heat. The sensation by which we dis- 
 tinguish between hot and cold bodies is, and must always have 
 been, a perfectly familiar one. The sense by which we per- 
 ceive it may be called the temperature sense. It does not seem 
 to be the same as the sense of touch, by which we distinguish 
 the forms of bodies. The sensations given by it depend upon 
 so many conditions that they are utterly untrustworthy as a 
 measure of the temperature of the body which we examine by 
 it. Thus the same mass of water will appear to the hand either 
 warm or cold, according as the hand has previously been im- 
 mersed in very cold or in very hot water; and two bodies of 
 different material, like wood and iron, which have been ex- 
 posed to the same conditions, and which all physical tests show 
 us must be at the same temperature, will not appear equally 
 hot or cold when tested by the hand. Men have always been 
 accustomed to speak of the cause of the sensation felt, when 
 we touch a hot body, as heat, and to assign the different sensa- 
 tions given us by the body in different circumstances to the 
 presence in it of more or less heat. Similarly the sensation 
 experienced when we touch a cold body has often been assigned 
 to the presence in the body of something called cold. But, 
 after all, the distinction between the sensation itself and its 
 cause was not very sharply drawn, and a great deal of con- 
 fusion exists in the early work on the subject beween the two 
 ideas of temperature and quantity of heat. Our sensations 
 immediately recognize temperature and changes of tempera- 
 ture, and attention was first turned to the study of those 
 physical relations of bodies which are connected with their 
 temperature. 
 
 Our object in the study of heat is to trace the sensation of 
 heat to its origin in material bodies, and to explain the 
 various phenomena exhibited by bodies in connection with 
 heat in terms of matter and the motions of matter.
 
 142 
 
 99. Thermometers. Our sensations being so uncertain in 
 their estimate of the temperature of bodies, it is of first im- 
 portance to obtain an instrument which will indicate tem- 
 perature. Such an instrument is called a thermometer. 
 
 The first thermometer was constructed by Galileo. It con- 
 sisted of a small glass globe fitted with a long tube. The open 
 end of the tube was inserted in water, and the globe was heated 
 until some of the air was expelled from it. When it was 
 allowed to cool, the pressure of the external air was greater 
 than that of the air in the globe, and so a column of water was 
 forced up the tube, until equilibrium was established. The 
 air thus shut off in the globe formed what we may call the 
 thermometric substance, and the top of the water column de- 
 termined its volume. In using this instrument, as in using 
 any thermometer, it was assumed, as a fundamental fact of 
 experience, that when two bodies, whose temperatures are 
 different, are brought into each others presence, their tem- 
 peratures will finally become equal. This equalization of tem- 
 perature is brought about by a lowering of the temperature of 
 the hotter body, and a raising of the temperature of the 
 colder body. It was also assumed that a change in the tem- 
 perature of a body is accompanied by a change in its volume, 
 and in particular that the volume of air increases as its tem- 
 perature rises. Starting with these fundamental principles, 
 the use of the thermometer as constructed by Galileo is evi- 
 dent. When it is kept in a room at the ordinary temperature, 
 the top of the water column will stand at a certain point in 
 the tube, which may be marked. If it is then transferred to a 
 hotter room, or if a hot body is placed around the globe, the 
 equalization of temperature already described will occur, and 
 the volume of the air in the instrument will increase. The 
 distance which the top of the water column is forced down 
 the tube is a measure of the change of temperature. For 
 various reasons, the principal one of which is the irregularity 
 introduced by changes in the external atmospheric pressure, 
 Galileo's thermometer will not give consistent indications of
 
 HEAT. 143 
 
 temperature. It was very goon superseded by instruments 
 constructed on a different plan. 
 
 Stimulated, no doubt, by this invention of Galileo, a body 
 of physicists, resident at Florence and united in a club called 
 the Accademia del Cimento, undertook the construction of 
 thermometers which should furnish satisfactory measures of 
 temperature. The instrument they made was similar in form 
 to the ordinary thermometers now in use. That is, it con- 
 sisted of a glass bulb joined to a graduated tube, the bulb and 
 part of the tube being filled with a liquid. The peculiarity of 
 their instrument consisted in this, that they attempted to 
 graduate the tube in such a way that the volume between two 
 marks of graduation was a fixed fractional part, generally 
 one-thousandth, of the volume of the bulb. The instrument 
 thus made was filled with alcohol, so that the top of the 
 column stood opposite one of the marks on the scale when the 
 instrument was exposed to some standard temperature. The 
 temperature chosen as standard was that of the air during the 
 first light frosts at the beginning of the winter. Their first 
 instruments were open at the top, but as this interfered with 
 their permanence and with their transportability, they were 
 afterwards closed. This plan of constructing thermometers 
 did not succeed in furnishing instruments which would give 
 similar indications at different temperatures, and notwith- 
 standing the partial success which was obtained much later 
 by Keaumur in the construction of thermometers on the same 
 plan, it has long ago been abandoned. 
 
 The method of graduating thermometers which is now uni- 
 versally used was describc-d by Dalence in 1688. and an instru- 
 ment made on that plan was constructed by Newton. In it no 
 attempt is made to establish any fixed relation between the 
 volumes indicated by the graduation and the volume of the 
 bulb. Only so much choice is exercised of the relative volumes 
 of the tube and the bulb as will ensure that the thermometer 
 can be used throughout the temperature range for which it is 
 intended. Newton used linseed oil as the thermometric sub- 
 stance. To graduate the instrument, he placed it first in a
 
 144 HEAT. 
 
 mixture of ice and water, the temperature of which waa 
 known to remain constant. He made a mark on the tube at 
 the point indicated by the top of the liquid column, after the 
 instrument had stood in the ice for some time, and showed no 
 signs of any further change. This point indicated one stan- 
 dard temperature. He then placed the bulb of the instrument 
 under his arm pit, and after the column again became station- 
 ary, he made another mark on the tube opposite the end of 
 the column. The temperature of the human body was known 
 to be very nearly constant, and the temperature thus obtained 
 was therefore taken by Newton as a second standard temper- 
 ature. The distance between the two standard marks thus 
 obtained was divided into twelve equal parts, and the gradu- 
 ation thus established was extended above and below the 
 standard marks. Newton did not choose the best thermo- 
 metric substance that can be used, or the most suitable stan- 
 dard temperatures, but the method which he employed was 
 essentially correct. It is easy to see that if two instruments, 
 in which the same thermometric substance is used, are gradu- 
 ated in this manner, they will not only indicate the standard 
 temperatures under the same conditions, but will agree in 
 their indications, to whatever temperature they are exposed. 
 The use of mercury as a thermometric substance was intro- 
 duced by Fahrenheit, to Avhom we owe the first thermometers 
 which compare with those now made in the accuracy of their 
 indications. Fahrenheit used the temperature of melting ice 
 as one of the standard temperatures, and probably the tem- 
 perature of boiling water under a standard pressure as the 
 other. It was subsequently discovered that the temperature 
 of boiling w r ater not only depends upon the atmospheric pres- 
 sure, but also upon the material of the vessel in which the 
 water is boiling. Cavendish therefore proposed to use as the 
 second standard temperature the temperature of steam over 
 boiling water, which -he showed to be independent of the 
 material of the vessel. It depends upon the atmospheric pres- 
 sure in a way which has been carefully determined by experi- 
 ment, so that a correction can be made to standard pressure.
 
 HEAT. 145 
 
 These two temperatures, the temperature of melting ice, and 
 the temperature of steam over boiling wfeter at a standard 
 pressure, are those now universally adopted as the two stan- 
 dard temperatures. 
 
 Fahrenheit assigned the number 32 to the first standard 
 mark, and the number 212 to the second standard mark. He 
 therefore divided the distance between the two marks, or as 
 we may call them, the freezing point and the boiling point, 
 into 180 parts or degrees. When this scale is extended below 
 the freezing point, the zero marks a temperature which is 
 very nearly that of a mixture of pounded ice and salt. 
 
 A change in the number of degrees between the freezing 
 point and the boiling point was recommended by Linnaeus and 
 carried out by Celsius. In the thermometers of Celsius there 
 were 100 degrees between the two standard temperatures. 
 The temperature of the freezing point was marked 0, so that 
 the temperature of the boiling point was 100. The scale thus 
 constructed is now generally called the Centigrade scale. It 
 is the one universally used in physical investigations, though 
 the Fahrenheit scale is still used in England and America by 
 meteorologists. 
 
 It should be observed that the scale of temperature which 
 has been adopted is a purely arbitrary one. The zero is an 
 arbitrarily chosen temperature, and the change of temperature 
 which is called a degree is determined by an arbitrarily chosen 
 change of volume of a standard substance. There is nothing 
 which tells us that the change of temperature which causes 
 this standard change of volume in one part of the scale is the 
 same as the change of temperature which will cause the same 
 change of volume in another part of the scale, it being under- 
 stood that by change of temperature in this statement is 
 meant a change measured by a change in the fundamental 
 physical condition of the body, which is the true measure of 
 its temperature. This thermometric scale, therefore, does not 
 furnish an absolute measure of temperature, and strictly 
 speaking only indicates differences of temperature.
 
 146 HEAT. 
 
 Many other arrangements are used for the measurement of 
 temperatures and temperature differences. They depend for 
 their operation upon some relation between some physical 
 property of a body which can be measured and its temperature. 
 Their indications are usually compared with those of a ther- 
 mometer, and are thus expressed in degrees. 
 
 100. Melting Temperatures, etc. Even with their imper- 
 fect thermometers the early observers made several important 
 discoveries in connection with the subject of temperature. The 
 Accademia del Cimento tried the experiment of immersing & 
 vessel filled with ice in a large mass of hot water, and observ- 
 ing the temperature of the ice with one of their thermometers. 
 They expected to find that the temperature of the ice would 
 fall, but found, in fact, that it remained constant until most 
 of the ice was melted. By trial with other masses of ice, 
 they found that ice always melts at appreciably the same tem- 
 perature. This temperature is called the melting point of ice. 
 Their observation was frequently confirmed by other observers, 
 and it was shown further that many other bodies possess 
 definite melting points, which are characteristic of the bodies. 
 By selecting a set of such bodies, melting at different tem- 
 peratures, a series of definite temperatures may be determined, 
 which are independent of the construction of any particular 
 instrument, and so far as we know, will be the same every- 
 where. The first scale of this sort was established by Newton. 
 
 Hooke discovered that, when water is boiled, its tempera- 
 ture is always approximately constant. This temperature, 
 called the boiling point of water, is not so independent of 
 external conditions as the freezing point is. In particular, it 
 depends upon the pressure in the vessel in which the water is 
 boiled. This had been proved some time before Hooke's dis- 
 covery by Boyle, who placed a vessel of water which was hot, 
 but not boiling, in the receiver of his air pump, and exhausted 
 the air from around it. When the exhaustion had reached a 
 certain point, the water began to boil, and it could be made to 
 boil again and again by exhausting the receiver still farther, 
 although it was continually cooling. On the other hand, if
 
 HKAT. 147 
 
 water is enclosed in a tight vessel, like a boiler, under high 
 pressure, it will not boil unless its temperature is raised far 
 above its boiling point in an open, vessel. 
 
 When water is boiled in an open vessel, it is under atmos- 
 pheric pressure, and this changes from time to time, so that 
 the boiling point of water, when tested by an accurate ther- 
 mometer, will not appear to be the same at all times. It will 
 be the same, however, if examined at times when the atmos- 
 pheric pressure is the same. In order to use the boiling point 
 as a standard temperature, the standard pressure of one atmos- 
 phere, or of 700 millimetres of mercury, has been selected as 
 the one at which the boiling point shall be standard. 
 
 101. Freezing Mixtures. The Accademia del Cimento dis- 
 covered several pairs of substances, which, when mixed with 
 each other, would produce very low temperatures. Such mix- 
 tures were called freezing mixtures, because the temperatures 
 which they produced were so low that other bodies could be 
 frozen by means of them. The mixture of ice and salt is a 
 familiar example. If a quantity of broken ice, at a tempera- 
 ture below the freezing point, while it is therefore a dry solid, 
 is mixed with salt, the temperature of the mixture falls until 
 it reaches Fahrenheit, or ahout 1H (.'enuirrade. Of course 
 it cools the bodies around it, and thus may be used to freeze a 
 liquid brought in contact with it. 
 
 It was noticed by Boyle, who devoted considerable atten- 
 tion to freezing mixtures, that the solid bodies which were 
 brought together in the mixture always melted, or, that if they 
 were melting already, they melted faster after being mixed. 
 Thus, in the mixture of dry ice and salt already described, both 
 ice and salt melt at a temperature below the melting point ot 
 either one of them. 
 
 102. Freezing Points. When water is exposed in a vessel, 
 for a sufficient time, to a temperature which is below its melt- 
 ing point, it will gradually freeze. While freezing, its tem- 
 perature remains constant at the melting point, that is, the 
 melting and freezing temperatures are the same. This fact 
 was not recognized by the earliest observers, because of the
 
 148 HKAT. 
 
 way in whicli the temperatures of different parts of a mass of 
 water differ on account of their differences in density, but it 
 was easily observed when small quantities of water were 
 rapidly frozen by means of freezing mixtures. The same gen- 
 eral statement holds true for other bodies which have definite 
 melting points, that their melting points and freezing points 
 are the same. 
 
 An apparent exception to this rule was discovered by 
 Fahrenheit. He showed that if a small quantity of water was 
 first boiled, so as to expel the air from it, and then allowed to 
 cool slowly in a smooth glass vessel, its temperature might fall 
 several degrees below the freezing point without its freezing. 
 The water in this condition is said to be supercooled. If 
 supercooled water is suddenly agitated, or if a grain of sand, 
 or better still, a crystal of ice, is dropped in it, it will at once 
 begin to freeze. Freezing goes on in this case much more 
 rapidly than when it begins at the freezing temperature. At 
 the same time, the temperature rises to the normal freezing 
 point. Very many liquids may be supercooled in a similar 
 way. and exhibit similar phenomena when they freeze. 
 
 103. Change of Volume on Freezing. When a liquid freezes 
 there generally occurs a rearrangement of its parts, such that 
 the density of the solid formed is different from that of the 
 liquid. Galileo noticed that this is true in the case 01 water 
 and ice, and showed, from the fact that ice floats in water, 
 that the density of the ice is less than that of the water. The 
 relative density of ice to water is about as 0.918 to 1. Metals 
 like bismuth, type metal, or even iron, with which sharp cast- 
 ings can be made, agree with water in having the density of 
 the solid state less than that of the liquid. In most cases the 
 change is in the opposite sense, and the density of the solid 
 is greater than that of the liquid. 
 
 104. The Temperature of Mixtures. Taylor, and after- 
 wards Riehmann, tried the experiment of mixing quantities of 
 water together whose temperatures were different, and observ- 
 ing the resulting temperature of the mixture. They found 
 that, when the two quantities of water were equal, the result-
 
 HEAT. 149 
 
 ing temperature was the mean of the original temperatures. 
 When tne quantities of water were not equal, this was not the 
 case. The resulting temperature was found to be given by 
 the following rule: Multiply the mass of each portion of 
 water by its temperature, add the products, and divide the 
 sum by the sum of the masses; the quotient is the resulting 
 temperature. This rule is known as Richmann's rule. 
 
 We may explain tne fact embodied in Richmann's rule, if 
 we suppose that the changes of temperature which occur in 
 the two masses of water are due to the passage of heat from 
 one to the other. That is, we suppose that, when the tem- 
 perature of water rises, it is because of the entrance into the 
 water of something which we call heat, and which we believe 
 to be the cause of temperature. If we assume that the 
 change of temperature produced by the mixture of the two 
 portions of water is due to the passage of heat from the hotter 
 portion into the colder, we find that Richmann's rule is con- 
 sistent with this assumption, if the quantity of heat which 
 enters a body is proportional to the mass of the body and to 
 the rise of temperature which occurs. For, Richmann's rule 
 is equivalent to the statement th'at the products of the masses 
 of water and their respective changes of temperature are 
 equal. 
 
 We have thus reached a conception of heat as something 
 which may enter or leave a body, which is distributed through- 
 out its mass, and which determines its temperature. We have 
 further found a way to measure it, at least to measure so 
 much of it as enters or leaves a body, by the observation of 
 the mass of the body and of its change of temperature. By 
 selecting a particular body and a particular change of tem- 
 perature, we may define a unit of heat. 
 
 A unit of heat which is frequently used in physical investi- 
 gations is called the calorie. It is the heat which will raise 
 the temperature of a kilogramme of pure water one degree 
 Centigrade. As recent observation has shown that the amount 
 of neat, measured in energy units, which will raise the tem- 
 perature of a kilogramme of water one degree, is slightly
 
 150 H KAT. 
 
 different in different parts of the scale, it is necessary, in order 
 to give greater precision to this definition, to specify the par- 
 ticular degree on the scale through which the temperature of 
 the water shall be raised. The degree usually chosen is that 
 between and 1 on the Centigrade scale, though other de- 
 grees have been chosen. It is often convenient to use a 
 smaller unit of heat than this, and we accordingly choose, as 
 another unit, the heat which will raise the temperature of a 
 gramme of water from to 1 Centigrade. This unit we may 
 call the gramme-degree. 
 
 105. Specific Heat. When masses of two different sub- 
 stances, whose temperatures are different, are mixed, the 
 resulting temperature does not conform to Bichmann's rule. 
 It was discovered by Black that the resulting temperature 
 may always be found, in a way consistent with the general 
 conception of heat which we fcave adopted, and by a rule 
 essentially similar to Richmann's, if we suppose that the 
 amount of heat required to raise the temperature of unit mass 
 of a particular substance one 'degree, is characteristic or spe- 
 cific for that substance. On this supposition, the quantity of 
 heat which a mass of a subst'ance will lose when its tempera- 
 ture falls a certain number of degrees, is equal to the specific 
 heat of the substance multiplied by its mass and by its 
 change of temperature. When portions of two different sub- 
 stances are mixed, so that heat passes from one to the other, 
 the resulting lemperature is found by a rule similar to Rich- 
 mann's rule if we replace the masses in Richmann's rule by 
 the products of the masses and the specific heats. 
 
 From the rule that has just been given, we may determine 
 the ratio of the specific heats of any two substances, by allow- 
 ing known masses of them, whose original temperatures differ 
 from each other, to exchange heat until they reach a common 
 temperature. If, in all our experiments, we use some one sub- 
 stance as standard, we may determine the ratio of the specific 
 heats of other substances to that of the standard substance, 
 and if we arbitrarily assign the value 1 to the specific heat of
 
 HKAT. 151 
 
 the standard substance, the ratios thus determined may be 
 called the specific heats of the other substances. 
 
 In the determination of the specific heats of solids and 
 liquids, the substance universally chosen as the standard is 
 water. The specific heat of any other substance is therefore 
 the ratio of the amount of heat which will raise the tempera- 
 ture of a mass of that substance one degree to the amount of 
 heat which will raise the temperature of an equal mass of 
 water one degree. If the mass considered in this definition is 
 a kilogramme, the ratio between the two quantities of heat is 
 the amount of heat, measured in calories, which will raise the 
 temperature of a kilogramme of the substance one degree. We 
 may therefore define the specific heat of a substance as the 
 amount of heat, measured in calories, which will raise the 
 temperature of a kilogramme of the substance one degree. 
 The heat capacity of a body is equal to the product of its mass 
 and its specific heat. 
 
 106. Latent Heat or Heat of Fusion. In the investigation 
 of specific heat, it was found by Black that the law by which 
 the resulting temperature of a mixture is determined does not 
 hold good if one of the bodies melts when it is brought in con- 
 tact with the other. In any such case, the resulting tempera- 
 ture is alway*s lower than that given by the law. Black was 
 therefore led to consider the process of melting, in order to 
 ascertain whether heat is required for it. In one of his ex- 
 periments he placed two similar vessels, one containing water, 
 the other, an equal mass of ice, on the top of a stove. The 
 temperature of the water at once began to rise. The ice in 
 the other vessel melted, but its temperature, and that of the 
 water which flowed from it, remained constant until melting 
 was complete. Then the temperature in that vessel also began 
 to rise, and rose at the same rate as in the other vessel. It 
 is plain that heat must have been entering both vessels all the 
 time and at the same rate, and since no evidence of its having 
 entered the ice was given by any change of temperature, it 
 must have been somehow used in melting the ice. In another 
 of his experiments, Black placed a mass of ice in a mass of
 
 152 HEAT. 
 
 warm water, and determined the temperature which resulted 
 when the ice was melted. On the supposition that the water 
 formed by the melting ice mixed with the warm water accord- 
 ing to Richmann's rule, the amount of heat which had appar- 
 ently disappeared ^rom the mixture could be calculated. It 
 was evidently used to melt the ice. The loss of heat in this 
 case is especially apparent when the experiment is tried in a 
 way indicated by Black. Suppose equal masses of ice and 
 water are taken, and the temperature of the water raised to 
 80 Centigrade. When the ice is immersed jin the water, it 
 begins to melt, and the temperature of the water falls more 
 and more as the melting proceeds, until just as the last trace 
 of ice disappears, the temperature of the mixtnre falls to 
 Centigrade. The warm water in this case has given up 80 
 calories, and this heat has occasioned no rise of temperature 
 in another body. We conclude that it has been used in melt- 
 ing the ice. 
 
 Black considered that when heat enters a body in such a 
 way that it causes a rise of temperature, it is in such a condi- 
 tion with relation to the body that it can be detected by the 
 temperature sense. He therefore called it sensible heat. On 
 the other hand, the heat which has passed into ice, and melted 
 it, cannot be detected by any change of temperature which it 
 causes, and is in a sense concealed in the water. He therefore 
 called it latent heat. This term is a very convenient one and 
 is often used, although our present conception of heat as a 
 form of energy makes it somewhat inappropriate. We may, 
 instead of it, use the term, heat of fusion. 
 
 The absorption of heat by melting ice is an example of what 
 occurs whenever a solid body melts. In every case of melting, 
 an amount of heat is absorbed, which depends upon the mass 
 of the body melted and upon the substance of which it is com- 
 posed. The amount- of heat, in calories, which is absorbed by a 
 kilogramme of a substance, when it melts, is called the latent 
 heat, or heat of fusion, of that substance. The heat of fusion 
 of ice is about 80 calories.
 
 HKAT. 153 
 
 Black perceived that when a body melts, even though its 
 melting is not brought about by the entrance of heat from 
 without, it will of necessity absorb heat, and thus, if it does 
 not receive heat from without, its own temperature will fall. 
 He explained in this way the behavior of freezing mixtures, 
 and showed that the general fact observed by Boyle, that all 
 such mixtures melt, is the one upon which their efficiency as 
 freezing mixtures depends. 
 
 Black also perceived that when a liquid freezes, it will give 
 out an amount of heat equal to that which was absorbed by it 
 when it was formed by melting. He explained in this way 
 the constant temperature of a liquid while it is freezing, and 
 also the rise of temperature which occurs when a supercooled 
 liquid begins to freeze. 
 
 Black also .studied the case of boiling liquids. From the 
 constancy of their boiling points he inferred that heat is ab- 
 sorbed by them during the process of boiling. The amount of 
 heat, in calories, required to turn a kilogramme of a liquid 
 into vapor at the boiling temperature is called the latent heat 
 of the vapor, or the heat of vaporization of the liquid. When 
 the vapor condenses again, it may be made to heat a quantity 
 of liquid of the same sort as that from which it is formed, and 
 thus to give evidence that heat is emitted by a vapor on con- 
 densation. By taking advantage of the equality between the 
 heat of vaporization and the heat emitted on condensation, the 
 heat of vaporization may be determined. 
 
 107. Calorimeters. A calorimeter is an instrument by 
 means of which a quantity of heat may be measured. 
 
 From the principles developed in the previous sections, it 
 is evident that we can measure the amount of heat which 
 leaves a body by the effect which it will produce in some other 
 body. It is not possible for us to measure all the heat which 
 a body contains. 
 
 The calorimeter used for the method of mixtures consists 
 of a vessel, isolated from surrounding bodies, so far as pos- 
 sible, so that the heat which is introduced into it will remain 
 in it without change. A known quantity of water is placed in
 
 154 HEAT. 
 
 this vessel, and its temperature is taken. If another body, say 
 a block of iron, of known mass, is raised to a known tempera- 
 ture, higher than that of the water, and if it is then trans- 
 ferred to the water, its temperature will fall and the tempera- 
 ture of the water will rise, until they have reached a common 
 value. This value is then determined. The amount of heat 
 lost by the iron in falling from its original temperature to 
 the common temperature, is equal to the amount of heat gained 
 by the water in rising from its original temperature to the 
 common temperature. The heat gained by the water is 
 measured in calories by the product of the mass of the water 
 and its change of temperature. Thus the heat lost by the iron 
 is determined. 
 
 A calorimeter employed by Black and by Wilcke consists 
 simply of a block of ice, in which a small cavity is made. 
 When the ice is at zero temperature, the interior of the cavity 
 is dry. To keep it so, it is covered with a slab of ice. The 
 body whose heat capacity is to be tested is heated to a known 
 temperature and transferred to the cavity. The heat which it 
 gives up will melt the ice around it. After its temperature 
 has fallen to that of the ice, and the ice no longer melts, the 
 water which has been formed is taken out and weighed. We 
 may state the result obtained in terms of an arbitrary unit 
 of heat, namely, the amount of heat required to melt a kilo- 
 gramme of ice. If this unit is used, the weight of the water 
 obtained, in kilogrammes, measures the amount of heat which 
 the body has lost. As we know that 80 calories are required 
 to melt a kilogramme of ice, it is easy to state this amount 
 of heat in calories. 
 
 This method of melting has been applied in several different 
 ways. The most ingenious of these was that invented by 
 Bunsen, who utilized th change of volume, which occurs when 
 a quantity of ice. melts, as a measure of the amount of ice 
 which was melted. 
 
 The method of condensation, which has been highly devel- 
 oped by Joly, measures the amount of heat, which will pro-
 
 HEAT. 155 
 
 duce a given change of temperature in a body, by weighing 
 the amount of steam which is condensed upon that body. 
 
 The method of cooling measures the amount of heat which 
 leaves a body by the rate at which its temperature falls. In 
 order to carry out the experiment on different substances 
 under similar conditions, an instrument is constructed con- 
 sisting of a small polished box, in the middle of which stands 
 the bulb of a thermometer. The box is first filled with a stan- 
 dard substance. It is then raised to a high temperature and 
 placed within a larger box, from which the air can be ex- 
 hausted, and whose walls are kept at a constant temperature 
 by immersion in ice. The thermometer is observed from 
 minute to minute, and the rate at which the temperature 
 changes for the standard substance is thus determined. Other 
 substances are compared with the standard by carrying out 
 similar observations with them. 
 
 The method of comparison depends upon the introduction 
 of equal quantities of heat, in any way in which that may be 
 done, into known masses of a standard substance and of the 
 substance under examination, and the observation of their 
 changes of temperature. 
 
 108. Specific Heats. Calorimetric observations are usually 
 employed to determine either the heat capacity of a body, or 
 the specific heat of a substance. The heat capacity of a body 
 is supposed to be constant within the range of temperature 
 employed in the experiment, and is therefore determined from 
 the quantity of heat which leaves the body, when its tempera- 
 ture undergoes a known change, by dividing that quantity of 
 heat by the change in temperature. . The specific heat of the 
 substance of which the body is composed, provided it is homo- 
 geneous, is obtained by dividing the body's heat capacity by 
 its mass. 
 
 By the study of the specific heats of solid substances 
 through different ranges of temperature, it is found that, as 
 a first approximation, they are constant at all ordinary tem- 
 peratures. As a rule, the specific heat of a substance increases 
 slightly as the temperature rises. There are a few substances,
 
 156 HKAT. 
 
 of which carbon, in the form of the diamond, is an example, 
 whose specific heat increases rapidly with rise of temperature. 
 The specific heat of the diamond is nearly three times as great 
 at 200 as at 0. 
 
 The specific heats of liquids also vary in a similar way 
 with the temperature. The most reliable observations indicate 
 a double variation in the case of water, its specific heat dimin- 
 ishing slightly from (J to about 35, and increasing from that 
 point on. 
 
 The specific heat of a substance in the liquid state is always 
 greater than that of the same substance in the solid state. 
 The specific heat of water, which has been taken as standard, 
 is greater than that of almost any other substance. So far 
 as known, the only specific heats which are greater than that 
 of watT are thusi- of hydrogen, and ut mixtures of water with 
 some <>f the alcohol-*. 
 
 The specific heats of a gas differ considerably according to 
 the circumstances in which the measurement is made. If the 
 gas is examined while its volume is kept constant, its specific 
 heat will have a certain A r alue, called its specific heat at con- 
 stant volume. If, on the other hand, it is examined while its 
 pressure is kept constant, so that as its temperature rises it 
 expands, it is found that an additional quantity of heat is 
 used in raising it to the same temperature, and its specific 
 heat is greater than in the other case. The specific heat thus 
 determined is called the specific heat at constant pressure. 
 Regnault proved, by direct observation, that the specific heat 
 of a gas at constant pressure is independent both of the pres- 
 sure and of the temperature of the gas. 
 
 A very remarkable relation among the specific heats of 
 those-chemical elements which are found in the solid state was 
 discovered by Dulong and Petit. These physicists, examining 
 the specific heats of thirteen of the solid elements by the 
 method of cooling, found that in each case the product of the 
 specific heat by the atomic weight of the element was approx- 
 imately the same number. Now, the masses of different ele- 
 ments which contain the same number of atoms are proper-
 
 HEAT. 157 
 
 tional to their atomic weights, so that the products of the 
 specific heats and the atomic weights are the heat capacities 
 of masses containing the same number of atoms, and since this 
 product is the same for many solid elements, we conclude that 
 for them their atoms all have the same capacity for heat. 
 This law was subsequently shown to apply approximately to 
 almost all the solid elements. The product here defined is 
 called the atomic heat of the element. Its value is about 6.4. 
 The atomic heats of a few of the solid elements, especially of 
 carbon and silicium, are exceptions to the general rule. 
 
 It was shown by F. Neumann and by Regnault that the 
 specific heats of substances which are compounds of the solid 
 elements are such. as to indicate that the atoms in composition 
 retain their atomic heats. That is, it is found that the pro- 
 ducts of the specific heats and the molecular weights of com- 
 pounds which have the same number of atoms in the molecule 
 are approximately equal, so that the molecular heats of such 
 compounds are equal. When the constituents of the molecules 
 are elements which conform to Dulong and Petit's law, the 
 quotient of the molecular heat divided by the number of 
 atoms in the molecule is found to be the constant atomic heat 
 already considered. When the molecule is one which contains 
 atoms of an element which cannot be examined in the solid 
 state, and other atoms of elements which conform to Dulong 
 and Petit's law, we may calculate the atomic heat of the un- 
 known element from the molecular heat of the compound. In 
 this way the atomic heat of the gaseous elements, when they 
 form parts of the molecules of solids, have been calculated. 
 It is thus found that the elements hydrogen, nitrogen, and 
 oxygen do not conform, at least in all cases, to Dulong and 
 Petit's law. 
 
 109. Transfer of Heat. It is a matter of common observa- 
 tion that heat may be transferred from one body to another. 
 Thus, when one end of an iron bar is thrust in the fire, the 
 other end gradually gets warmer, and a body may be warmed 
 by placing it in front of the fire, although no part of it is in 
 the fire. An experiment described by Newton proves that in
 
 158 HKAT. 
 
 the latter case the heat is not transferred from the fire to the 
 body by the action of any known material body between them. 
 Newton placed two thermometers in two similar glass vessels, 
 from one of which the air was exhausted. After letting them 
 stand in a cool place until the thermometers indicated the 
 same temperature, he transferred them to a warm place, and 
 found that the temperature of the thermometer in the vacuum 
 rose nearly as fast as that of the other one, and that the final 
 temperatures of both were the same. The heat which reached 
 the thermometer in the vticuum was manifestly translerred to 
 it from the walls of the vessel, when they were heated by 
 standing in the warm place, and since there was no known 
 material medium in contact with the bulb of the thermometer, 
 the heat which it received must have been transferred to it 
 without the intervention of any such medium. 
 
 Heat transferred in this way is said to be transferred by 
 radiation, and is called radiant heat. Its properties are in 
 every respect like those of light. Indeed, subsequent study 
 has proved that radiant heat and light are essentially similar 
 in all respects, arid we shall therefore study it in connection 
 with light. One general principle was discovered, however, 
 governing the radiation of heat, which does not depend on the 
 mode in which it is propagated, and which may be considered 
 in this place. 
 
 Taking advantage of the fact that radiant heat may be re- 
 flected and brought to a focus, as light Is, the following experi- 
 ment was tried: Two spherical mirrors were set up facing 
 each other. A thermometer was placed at the focus of one of 
 them. At the focus of the other was placed a ball of lead 
 which had been heated, though not to redness. When this was 
 done, the temperature of the thermometer at the other focus 
 began to rise, showing a reception of heat by the thermometer 
 from the hot lead.- When a block of ice was substituted for 
 the lead, the temperature of the thermometer fell. Consider- 
 ing these experiments, Prevost perceived that the rational way 
 to explain the fall of the thermometer produced by the ice, 
 was to ascribe it, not to a radiation of cold from the ice, but
 
 HEAT. 159 
 
 to a radiation of heat from the thermometer. Generalizing 
 this idea, he laid down the principle that all bodies are at all 
 times radiating heat, and receiving heat from neighboring 
 bodies, and that; the change of temperature of the body de- 
 pends upon the relative amounts of heat which it is receiving 
 and emitting. When it receives more than it emits, its tem- 
 perature rises; when it emits more than it receives, its tem- 
 perature falls; when it emits the same amount as it receives, 
 its temperature is constant. This principle is known as Pre- 
 vost's law of exchanges. 
 
 Heat which is transferred through a solid, when one part 
 of it is heated, is said to be transferred by conduction. The 
 experiments of Richmann and of Ingenhouss showed that the 
 rate at which heat is transferred by conduction is different in 
 different substances. It will manifestly depend also on the 
 differences of temperature in the body, or on the way in which 
 the temperature of the body changes- along the lines along 
 which conduction takes place. By assuming that the flow of 
 heat along a line is proportional to the rate of change of tem- 
 perature along that line, Biot, Rumford, and subsequently 
 Fourier, were able to explain the movement of heat in bodies 
 in a way which is consistent with the results of observation. 
 
 In accordance with the foregoing assumption, we may de- 
 fine the conductivity of a substance as the amount of heat 
 which, in a unit of time, will pass between two unit areas in 
 the substance, so placed that they stand at unit distance 
 apart and that between them the temperature differs by one 
 degree. 
 
 When heat passes across a surface at which two substances 
 meet, its rate of transfer, or its surface conductivity, depends 
 on the nature of the substances. As a first approximation, it 
 is assumed to be proportional to the difference of temperature 
 between the two substances. We may define the surface con- 
 ductivity as the amount of heat which will pass, in one unit 
 of time, through unit area of the surface, when the difference 
 of the temperatures on the two sides of the surface is one 
 degree.
 
 160 HEAT. 
 
 In most cases in which bodies transmit heat by conduction, 
 the temperatures of the different parts of the body will grad- 
 ually approach definite values. After these definite values are 
 attained, no more temperature changes occur. In all such 
 cases it is plain that each part of the body receives as much 
 heat from the hotter portions of the body as it sends on to 
 the cooler portions. This condition of the body is called its 
 steady state. In many other cases, in which the source of 
 heat is not applied to the body continuously, the temperatures 
 of its different parts vary continually in a way which depends 
 on the way in which the heat is applied, on the shape and size 
 of the body, and on its conductivities. Many such cases have 
 been studied by the help of the assumptions already describe;!, 
 and the theoretical results obtained have been found to agree 
 with the results of observation. 
 
 The transfer of heat in a liquid takes place generally by a 
 process which is known as convection. It is a well-known fact 
 that liquids receive heat readily. This was ascribed at first 
 to liquids being very good conductors. Rumford noticed, how- 
 ever, that masses of liquid suspended in fibrous bodies take 
 up heat very slowly, and retain it for a long time. This ob- 
 servation seemed to him inconsistent with the hypothesis that 
 liquids are good conductors, and he accordingly undertook an 
 investigation of the behavior of liquids when heated. Taking 
 a glass flask with a long neck, he filled it with water, in which 
 were suspended small particles or motes, and set it in a room 
 whose temperature was low and constant. When it had stood 
 there until its temperature had become that of the room, and 
 until the particles suspended in the water were still, showing 
 that there were no currents in the water, he transferred it to 
 a warm room. The suspended particles at once began to move 
 upward along the walls and to descend in the middle of the" 
 flask, showing that a regular circulation of the water was 
 occurring. This circulation kept up until the temperature of 
 the water had risen to that of the room. When he transferred 
 the flask to the cold room again, currents in the opposite sense 
 occurred, the water moving downward along the walls and
 
 HKAT. 161 
 
 upward in the middle. After these currents were once per- 
 ceived, it was easy to explain them. When the flask was 
 brought into the warm room, its walls took up heat from the 
 surrounding air, and heated the layers of water which were 
 near them. The density of the water was diminished by its 
 expansion, due to this heating, and it therefore rose along the 
 walls. The denser, because cooler, portions of the water which 
 sank to give room at the top for the warmer portions, came 
 in turn in contact with the walls, were also heated, and rose 
 along the walls. In this way are explained the rapid reception 
 of heat by a liquid, and the fact that the temperature of a 
 liquid which is being heated from the bottom is almost the 
 same throughout. 
 
 To test whether a liquid is a good conductor in the ordi- 
 nary sense, Rumford tried to heat a mass of water by apply- 
 ing the heat at the top, so that the usual convection currents 
 could not arise. He found so little heat transmitted to the 
 bottom of the vessel, that he concluded that water did not 
 conduct at all. This conclusion is manifestly erroneous, for 
 if there were no conduction in water, the heating of the water 
 near the walls of the flask in Rumford's first experiment, could 
 not be accounted for. As was shown by subsequent observa- 
 tion, water and other liquids are not non-conductors of heat, 
 but very poor conductors. 
 
 Gases are ordinarily heated by convection. It was shown 
 by Magnus that gases, like liquids, are poor conductors. 
 
 110. Expansion of Solids. The general truth that a body 
 expands when its temperature rises was illustrated in the 
 early work on the subject of heat, by the expansion of the air 
 in Galileo's thermometer, and by the behavior of thermometric 
 substances in general. The Accademia del Cimento demon- 
 strated that metals expand when they are heated. After the 
 discovery of the general truth, it became a matter of interest 
 to investigate the laws of this expansion. 'An 5nstrument 
 called the pyrometer was invented for that purpose, which 
 consisted essentially of a frame work so arranged that one 
 end of the bar of metal under investigation could be kept
 
 162 11 BAT. 
 
 fixed, while the other end was attached to a rack and pinion, 
 by which a pointer could be moved over a dial. The rod was 
 immersed in a vessel full of water or oil, which was heated 
 from beneath. If the temperature of the water was raised and 
 the bar expanded, the pointer moved around the dial, and thus 
 measured the elongation of the bar. Inaccurate as this ar- 
 rangement was, it served to show that the elongation of a bar 
 of given length is proportional to the rise of temperature 
 which occasions it, and that bars of different metals have dif- 
 ferent elongations for the same rise of temperature. This lat- 
 ter statement was illustrated by an experiment made by 
 DeLuc, who clamped two bars, of iron and brass, lirmly to- 
 gether at one end, and observed the relative expansion, when 
 the bars were heated, by measuring the way in which the 
 length of one bar increased more than the length of the other. 
 The first accurate observations of the absolute elongation 
 of a bar were made by Laplace and Lavoisier. In their experi- 
 ments, one end of the bar rested against a massive stone pier. 
 The other end engaged with the short arm of a lever or sys- 
 tem of levers, on the last long arm of which a mirror was 
 mounted. While the bar was at the temperature of melting 
 ice, the image of a vertical scale, reflected in this mirror, was 
 observed by a telescope. The bar was then raised to the tem- 
 perature of boiling water. The consequent elongation moved 
 the levers, and so changed the position of the mirror. By 
 another observation of the scale, and from a knowledge of the 
 ratios of the arms of the levers, it was possible to calculate 
 the elongation of the bar. It was found that the elongation 
 of unit length of the bar, produced by different changes of 
 temperature, was not proportional to the change of tempera- 
 ture in each case. To represent the length of fhe bar at dif- 
 ferent temperatures an expression involving the second, or 
 even the third, power of the temperature had to be employed. 
 It is however true, as a first approximation, that the elonga- 
 tion is proportional to the rise of temperature. If we confine 
 ourselves to this approximation, we may define the coefficient
 
 of expansion of a solid as the elongation of unit length of it 
 when it?" temperature rises Irom to 1 Centigrade. 
 
 For most practical purposes this approximation is all that 
 is needed. There is, however, one important operation, 
 namely, the construction and comparison of standards of 
 length, which requires an accurate knowledge of the co- 
 efficients of expansion of the bars of metal on which those 
 standards are marked off. 
 
 111. Expansion of Liquids. When a liquid expands by 
 heat in a thermometer bulb or in any similar vessel, the ex- 
 pansion which is observed by the rise of the column is the 
 relative expansion of the liquid and the vessel. That is, the 
 volume of the vessel increases as well as the volume of the 
 liquid, and the change of volume indicated by the rise of the 
 column is the difference between these two changes of volume. 
 The absolute expansion, which it is sometimes important to 
 determine, is the actual increase in volume of the liquid. 
 DeLuc observed the relative expansions of various liquids and 
 glass, and compared their absolute expansions by observing 
 the rise of liquid columns in similar thermometer tubes. In 
 the course of his observations he discovered a remarkable fact 
 in the case of water. As the temperature of the water ther- 
 mometer rose from the temperature of melting ice, the column 
 in the tube at first fell, showing a contraction of the water. 
 At 5 CVnti-rrHde. according to hi* observations, the column 
 readied its lowest point, and from that temperature on it 
 rose, showing a regular expansion. This observation has been 
 repeatedly confirmed. The temperature at which the volume 
 of the water is least, or at which its density is greatest, is 
 really 4 Centigrade. This fact, together with tho fact that 
 ice is less dense than water, plays an important part in the 
 economy of nature; for it is on that account that the temper- 
 ature of the water in large ponds and lakes rarely falls below 
 4, except near the top. 
 
 The most accurate study of the expansion of liquids has 
 been made with the hydrostatic balance, by determining the 
 apparent loss of weight of a standard body, like a hollow glass
 
 164 HEAT 
 
 sphere, when immersed in the liquid at different temperatures. 
 If the coefficient of expansion of the standard body is known, 
 the coefficient of expansion of the liquid may be calculated 
 from such observations. The coefficient of expansion in this 
 case is denned as the increase in volume of unit volume of the 
 liquid, when its temperature rises fn-m i<> 1 (.'entiiiniile. 
 The volume calculated by this coefficient for any temperature 
 will only be approximately correct. For accurate results, a 
 more complicated temperature function must be used. 
 
 112. Expansion of Gases. The first study of the expansion 
 of gases, or of what amounts to the same thing, their increase 
 in pressure with rise of temperature, was made by Amontons. 
 His instrument we may call an air-pressure thermometer. It 
 was a glass globe, into the bottom of which was inserted the 
 short limb of a recurved tube. Mercury wag introduced into 
 the globe until it was about half filled, and so that the top of 
 the column in the long limb of the tube stood at the same level 
 as that of the mercury in the globe, when the instrument was 
 at the standard temperature of melting ice. When this in- 
 strument was exposed to a higher temperature, the effect of 
 the expansion of the enclosed air was to force down the mer- 
 cury in the globe, and so to elevate the mercury column in tin- 
 tube. Since the area of the mercury surface in the globe was 
 very many times greater than that of the cross section of the 
 tube, the elevation of the mercury in the tube was as many 
 times greater than the depression of the mercury in the globe. 
 The air in the globe was thus subjected to pressure, propor- 
 tional to the elevation of the mercury column, and its volume 
 was maintained almost unchanged. If it had been kept ex- 
 actly constant by a further increase of pressure, produced by 
 adding more mercury to the column, the instrument would 
 have been a perfect air-pressure thermometer. As it was, it 
 served very well to enable Amontons to determine approxi- 
 mately the relation between the rise of temperature and the 
 consequent increase in pressure. He stated that when the 
 temperature rose from that of melting ice to that of boiling
 
 HEAT. 165 
 
 water, the increase in pressure was one-third the pressure at 
 the lower temperature. 
 
 The later attempts which were made to obtain a measure 
 of the expansion of gases, were for a long time failures. It 
 was shown by Dalton and by Gay-Lussac that these failures 
 could be traced to the presence of water vapor, or rather of 
 water, in the vessel containing the gas. By making the in- 
 terior of the vessel perfectly dry, and by drying the gas, Gay- 
 Lussac at last made a successful study of the expansion of 
 gases. He stated his results as follows: 
 
 1. All gases, whatever be their density, and all vapors, ex- 
 pand equally for the same change of temperature. 
 
 2. For the permanent gases the increase from the ice point 
 
 I f\r\ 
 
 to the boiling point is of the original volume 
 
 '266.66 
 
 The general law embodied in these statements is known as 
 Gay-Lussac's law. The original factor given in Gay-Lussac's 
 statement of the law has been slightly modified by subsequent 
 observations. We now know that the increase in volume of a 
 gas when its temperature rises from the melting point to the 
 
 1 00 
 
 boiling point is . of the original volume. The coefficient of 
 
 273 
 
 expansion of all gasf>s is therefore - . This statement is 
 
 closely accurate for those gases which can be condensed only 
 with great difficulty. Those which are easily condensed have 
 generally higher values of the coefficient. 
 
 In much of our study of gases, we consider a gas called the 
 ideal gas, which has no precise counterpart in nature. It is 
 defined to be a gas which obeys Boyle's and Gay-Lussac's laws 
 exactly. Consider such an ideal gas confined in a vessel of 
 volume v under the pressure p, t the temperature 0. If the 
 temperature is lowered one degree, the volume will diminish 
 
 by -_ part of the volume at zero. If the temperature falls 
 273 
 
 two degree?, the diminution of volume will be __ parts of the 
 
 273
 
 106 IIKAT. 
 
 volume at zero; and so on, as the temperature fulls lower an I 
 lower. When the temperature falls to 273 degrees, the volume 
 of the gas will vanish. 
 
 This consequence of Gay-Lussac's law is not so easily con- 
 ceived as an equivalent result obtained by supposing the 
 volume of the gas to be kept constant, and the pressure to 
 change because of the fall of temperature. As the temperature 
 
 falls, the pressure diminishes by losing - part of the press- 
 
 '27$ 
 
 Ire at for each fall of one degree, so that when the tem- 
 perature has fallen 273 degrees, the gas will no longer exert 
 pressure. This behavior of an ideal gas gives a hint that there 
 may be a temperature at which an ideal physical body will lose 
 some, at least, of its physical characteristics. We have shown 
 that the pressure of a gas may be explained by ascribing it to 
 the movements of the molecules which compose the gas. If 
 we accept this theory as the true explanation of the properties 
 of a gas, it follows at once that, when a gas exerts no pressure, 
 its molecules are at rest. On this theory, therefore, the tem- 
 perature 273 is the temperature at which the molecules of 
 an ideal gas would cease to move. 
 
 We know from Boyle's law that the pressure of a gas is 
 inversely proportional to its volume, and from Gay-Lussac's 
 law that the pressure is directly proportional to the factor 
 
 4- at, in which a= -^ is the coefficient of expansion and t 
 
 '2 1 3 
 
 is the temperature on the Centigrade scale. We may combine 
 these proportions into an equation by introducing a constant 
 
 or factor of proportion, so as to have p = c a If we 
 
 introduce in this equation the numerical value of the coefficient 
 
 of expansion, we obtain from it the equation pv= [: (273-H). 
 
 2 1 "i 
 
 Now e is a constant, to which we give the symbol R; and 
 2(3 
 
 the quantity in parenthesis is the temperature on the Centi- 
 grade scale increased by 273. If, therefore, we suppose a
 
 HEAT 167 
 
 thermometer so graduated that its zero indicates the tempera- 
 ture of 273 Centigrade, and that the length of its degree is 
 the same as that of the Centigrade degree, the temperature 
 indicated by Centigrade will be indicated by 273 on the 
 new scale, and any other temperature indicated by * on the 
 Centigrade scale, will be indicated by 273-t-t on the new scale. 
 We designate temperature given in the new scale by T. Tn 
 terms of this new notation, we have the equation pv=RT, as 
 a statement of Boyle's and Gay-Lussac's laws. 
 
 In the kinetic theory of gases we showed that the pressure 
 of a gas whose volume is kept constant is proportional to the 
 mean kinetic energy of its molecules. From the equation just 
 obtained, it therefore follows that the temperature of a gas, 
 measured on this new scale, is proportional to the mean kinetic 
 energy of its molecules. Now the least value which kinetic 
 energy may have is zero, and it has no negative values. This 
 relation therefore indicates that there can be no temperatures 
 lower than the temperature indicated by the zero of the new 
 scale, and that this zero is a limit of falling temperature. For 
 reasons which will subsequently be given, it is called the abso- 
 lute zero, and the scale, formed as we have described, is called 
 the absolute scale of temperature. We have no warrant at 
 present for the use of such terms. We shall accordingly call 
 this zero, the zero of the ideal gas thermometer, and the scale, 
 the scale of the ideal gas thermometer. 
 
 A thermometer which will indicate this scale very nearly 
 may be made by enclosing a mass of dry air in a vessel by 
 means of a mercury column, so arranged that, as the volume 
 of the air changes, the pressure on it may be kept constant. 
 On this condition, changes in temperature will be proportional 
 to the changes of volume. Such an arrangement, however, is 
 not so satisfactory in its working as one in which the volume 
 of the air is kept constant, as its temperature changes, by 
 suitable changes of pressure. With this instrument the tem- 
 peratures are taken proportional to the pressures, so that if 
 the pressure on the air is determined for a known tempera-
 
 168 HEAT. 
 
 ture, like that of melting ice, any other temperature may be 
 obtained by an observation of the corresponding pressure. 
 
 A property of gases which is connected with the expansion 
 with rise of temperature was discovered by Erasmus Darwin 
 and investigated by Dalton. They found that when air, or 
 any other gas, which has been under pressure in a closed ves- 
 sel, is allowed to expand suddenly into the atmosphere, it is 
 cooled by the expansion, and that if it is suddenly compressed 
 it will be heated. The heating produced by compression may 
 be shown with the so-called lire-syringe, the invention of a 
 French workman. 
 
 Another and more important general property of gases was 
 investigated by Gay-Lussac, and more satisfactorily by Joule. 
 Joule's experiment required the use of two stout metallic 
 cylinders, in one of which air was compressed under a high 
 pressure. From the other the air was exhausted. The two 
 cylinders thus prepared were connected by a tube, fitted with 
 a stop-cock, which could be opened to permit the compressed 
 air to rush into the empty cylinder. They were immersed in 
 water in a large calorimeter, and the temperature of the water 
 observed. The stop-cock was then opened and the air was 
 allowed to distribute itself uniformly in the two cylinders. 
 Another observation of the temperature of the water showed 
 no change in it. As the air had passed into the empty cylin- 
 der without doing external work, its energy remained the 
 same in the larger volume as it had been in the smaller one. 
 Joule concluded that the energy of a gas does not depend on 
 its volume. Since the temperature of the air did not change, 
 we know from Boyle's law that the produ-ct of its pressure 
 and volume did not change. We also know from the kinetic 
 theory of gases that this product is proportional to the mean 
 kinetic energy of the molecules of air. Since this mean kinetic 
 energy remains constant, we infer that it alone is the energy 
 of the gas. Now, the kinetic energy of the molecules, and 
 therefore also the energy of the gas, changes when the tem- 
 perature changes. We may therefore state the result obtained
 
 by Joule in the law, that the energy of an ideal gas is a func- 
 tion of its temperature, but is independent of its volume. 
 
 Later experiments by Joule and Lord Kelvin showed that 
 a slight fall of temperature usually occurs when a gas ex- 
 pands without doing external work. The law stated holds for 
 the ideal gas. 
 
 113. Vapors. If water is exposed in an open vessel to the 
 air it will gradually disappear. It is said to have evaporated, 
 and the process by which it disappears is called evaporation. 
 Many other liquids evaporate as readily as water does, or even 
 more readily. On the other hand, there are many liquids, like 
 sulphuric acid or mercury, which evaporate so slowly that 
 their evaporation can hardly be detected. If the temperature 
 of the water is observed while it is evaporating, it will be 
 found to be lower than that of the surrounding air. If the 
 water is placed in the receiver of an air pump, and the air 
 around it exhausted, evaporation will go on much more rapidly 
 than under atmospheric pressure, and the difference of tem- 
 perature between the water and surrounding bodies will be 
 much greater. During the process heat is entering the water 
 from without, and the rate at which it enters depends on the 
 difference of temperature between the water and surrounding 
 bodies. We therefore infer that the process of evaporation 
 involves the absorption of heat, and that the amount of water 
 evaporated at a given temperature is proportional to the 
 amoxint of heat which enters it. This conclusion has been 
 confirmed by many observations. 
 
 The result of the evaporation of water is the production of 
 another body, which is. called the vapor of water, or water 
 vapor. In general it seems to be similar to a gas. We may 
 study the properties of a vapor by inserting a small quantity 
 of the liquid, whose vapor we wish to examine, under the open 
 end of a filled barometer tube. The liquid will rise through 
 the mercury of the column, and will evaporate in the vacuum 
 above. Ordinarily, when the space above is not too great, 
 some of the liquid will remain unevaporated and will float on 
 top of the mercury column. The first thing to be noticed is
 
 that, as soon as the vapor is formed, the mercury column is 
 depressed. This indicates that the vapor is exerting a pres- 
 sure upon the top of the column. The change in height of the 
 mercury column is a measure of this pressure. If the bar- 
 ometer tube stands in 30 deep a vessel, and is itself so long, 
 that we may raise or lower it so as to make considerable 
 changes in the volume of the space above the column, we find 
 on changing the volume, that the pressure indicated by the 
 height of the column above the mercury surface outside the 
 tube remains constant. This is not the way in which a gas 
 would behave, for its pressure changes when its volume 
 changes. If so little liquid has been introduced into the tube 
 that none of it remains unevaporated, the pressure will be 
 less than that obtained when liquid is present, and it will not 
 remain constant when the volume is changed. The vapor in 
 this case will behave like a gas. It is only when liquid as well 
 as vapor are present in the tube that the constant and maxi- 
 mum value of the vapor pressure is exhibited. 
 
 If the instrument, arranged as described, is exposed to a 
 higher temperature, the top of the mercury column will be 
 depressed, showing an increase in the vapor pressure. On the 
 other hand, a cooling of the instrument will show a decrease 
 in the vapor pressure. The rate of increase or decrease is 
 different for different liquids, and has not been found to be 
 expressible in any general or simple law. 
 
 A vapor in the condition in which it exhibits its maximum 
 pressure is called a saturated vapor. We may restate the ex- 
 perimental results already described by saying that the press- 
 ure of a saturated vapor is a function of its temperature only, 
 being independent of the volume occupied by the vapor. We 
 may explain this law by supposing that a saturated vapor is 
 one in which every unit of volume contains as much vapor TS 
 can exist in it at the given temperature without condensation. 
 If the volume of the vapor is diminished, enough of the vapor 
 condenses into the liquid state to keep the density of the 
 remaining vapor the same as before. If the volume is in-
 
 IIKAT 171 
 
 creased, enough of the liquid evaporates to saturate the larger 
 volume. 
 
 When the volume of the vapor is kept fixed, while its tem- 
 perature is raised, the increase in pressure exhibited by it is 
 not due simply to the increased energy of its molecules, but 
 also to the fact that more of the liquid evaporates, so that the 
 density of the vapor is increased. On the other hand, when 
 the temperature is lowered, some of the vapor is condensed. 
 
 It was discovered by Dalton that when vapors of two dif- 
 ferent liquids are formed in the same volume, the maximum 
 pressure which they exert is equal to the sum of the maximum 
 pressures exerted by the two vapors separately. The same 
 thing is true for mixtures of several vapors, or, if the volume 
 is kept constant, for mixtures of gases, or of vapors with gases. 
 In all these cases, the pressure of the mixture is equal to the 
 sum of the pressures of its constituents, if they were to occupy 
 the same volume separately. This law T is known as Dalton's 
 law. 
 
 The evaporation which has been described .takes place at 
 the free surface of the liquid. Another mode of evaporation 
 occurs, called ebullition or boiling, in which the evaporation 
 occurs within the body of the liquid. When the temperature 
 of the liquid has reached a certain point, which depends upon 
 the pressure upon it, bubbles of vapor appear in it, arising at 
 some point on the wall of the containing vessel. After boiling 
 has fairly begun, these bubbles rise through the liquid, rapidly 
 increasing in volume as they do so, and break at the top, 
 liberating the vapor which they contain. The temperature at 
 which this process takes place generally depends, for a given 
 liquid, on the pressure upon it, but it has been shown, by 
 Dufour, that if the liquid is prepared by previous boiling *o 
 that the air ordinarily dissolved in it is driven out, its tem- 
 perature may be raised considerably above its ordinary boiling 
 point, without its boiling. When boiling logins in this case, 
 the formation of vapor is exceedingly rapid. These phenom- 
 ena are analogous to those observed when a liquid is super- 
 cool ed*.
 
 172 HKAT. 
 
 The striking similarity of vapors and gases suggests that 
 it may be possible to explain the properties of vapors by the 
 kinetic theory. In order to do so, we extend the kinetic 
 hypothesis to liquids, so far as to assume that the molecules 
 of a liquid are in motion, and that they are not all moving 
 with the same velocity. Their average velocity will have u. 
 value depending on the temperature, but the velocities of the 
 separate molecules may, in some cases, very much exceed this 
 average velocity. We apply this hypothesis to a liquid exposed 
 in a closed and otherwise empty vessel. Among the molecules 
 of the liquid, which are moving in various directions, there 
 will be some near the upper surface which are moving upward, 
 and some of these will be moving with velocities which are 
 high enough to carry them beyond the attraction of the neigh- 
 boring molecules of the liquid. They then enter the space above 
 as molecules of vapor, and in this way the vapor is formed. 
 Now the molecules of the vapor are also moving in various 
 directions, and some of them will come within the range of 
 the forces of the liquid, and will return to it. The vapor will 
 attain its greatest density when it is so dense that the mole- 
 cules which enter the liquid from the vapor are equal in 
 number to those which enter the vapor from the liquid. The 
 attainment of this condition plainly does not depend on the 
 total number of molecules in the vapor, but only on the density 
 of the vapor just above the liquid, and it therefore follows 
 that the maximum pressure of the vapor is independent of its 
 volume. Since the velocity of the molecules increases as the 
 temperature rises, there will be more molecules at the higher 
 temperature whose velocities will be sufficient to ' carry them 
 awky from the liquid, and consequently the density of the 
 vapor above the liquid will have to be greater, at the higher 
 temperature, in order that as many molecules may leave the 
 vapor as enter it. Therefore the density and the pressure of 
 a saturated vapor will increase on rise of temperature. 
 
 114. Critical Temperature. Caignard de la Tour tried the 
 experiment of heating ether, when it was sealed up in a strong 
 glass tube. Under these conditions, as the temperature* rose,
 
 HEAT. 173 
 
 the vapor in the tube became more dense. The surface of 
 separation between the liquid below and the vapor above, called 
 generally the meniscus, remained distinctly visible until a cer- 
 tain temperature was reached. At that temperature it dis- 
 appeared, and the contents of the tube became apparently 
 homogeneous. Before the disappearance of the meniscus, its 
 position in the tube indicated the presence of a considerable 
 mass of liquid, and its disappearance did not seem to be due 
 to gradual evaporation, but to the attainment of a condition 
 in which the liquid did not dift'er in appearance from the vapor 
 above it. We may anticipate the discussion which is to follow 
 so far as to name the temperature at which this change occurs, 
 the critical temperature. For higher temperatures the con- 
 tents of the tube seemed homogeneous. 
 
 When the tube was cooled again, and the critical tempera- 
 ture was reached, a sudden condensation, in the form of a fine 
 fog or rain, took place throughout the tube, and the liquid 
 reappeared. 
 
 Similar experiments were tried with other liquids, and for 
 many of them the existence of a critical temperature was de- 
 termined. 
 
 If a non-saturated vapor is compressed into a small volume, 
 it will at last reach the condition of saturation, and condensa- 
 tion will begin. This fact suggested the possibility of con- 
 densing the gases into liquids by increasing the pressure upon 
 them. For some time attempts made in this direction were un- 
 successful. Although the gases in some cases were subjected 
 to enormous pressures, Natterer, for example, using pressures 
 as high as three thousand atmospheres, they showed no signs 
 of liquefaction. The first successful experiments were made by 
 Faraday. The method which he employed involved the use of 
 a strong glass tube, bent in the middle. Substances which, by 
 their chemical action, would produce the gas to be examined, 
 were introduced in one of the limbs of this tube, and the tube 
 was then sealed. The gas being continually generated, the 
 pressure on it increased. Taking a hint from de la Tour's 
 observations, Faraday surrounded the end of the tube which
 
 1 74 H EAT. 
 
 did not contain the substances generating the gas with u 
 freezing mixture, in order, if possible, to lower the temperature 
 of that part of the tube below the critical temperature. With 
 this arrangement he liquified chlorine, carbon dioxide, and 
 many other gases. A few of the gases, among them hydrogen, 
 nitrogen and oxygen, proved refractory to this treatment. 
 Faraday recognized that the reason of this was his inability 
 to obtain a temperature as low as the critical temperatures of 
 these gases. 
 
 This notion of a critical temperature marking the highest 
 temperature at which a gas or vapor can be condensed into a 
 liquid and marking a limit above which a gas or vapor cannot 
 be condensed into a liquid by any pressure whatever, was 
 developed by Andrews in connection with his experiments on 
 the behavior, of carbon dioxide under pressure. 
 
 The refractory gases, or at least some of them, were con- 
 densed by Cailletet and by Pictet. The methods used by these 
 investigators were essentially the same. The gas to be con- 
 densed was forced under high pressure into a small chamber 
 furnished with a stopcock, opening to the air. After it had 
 been cooled to the lowest attainable temperature, the stopcock 
 was opened, and it was allowed to rush out. The additional 
 cooling, produced by the sudden expansion, brought the tem- 
 perature down below the critical value, and a little of the 
 gas condensed as a liquid, under atmospheric pressure, on the 
 walls of the tube. 
 
 By using more powerful refrigerating agents, Olszewski, 
 Dewar and others have succeeded in lowering the temperature 
 below the critical value, while the gas is under pressure, and 
 in this way have obtained large quantities of nitrogen, oxy- 
 gen and other gases in the liquid state. By allowing the liquid 
 thus obtained to evaporate rapidly, its temperature is still 
 further lowered. In this way nitrogen has been frozen at 
 220 Centigrade. The lowest temperature obtained in these 
 experiments is> estimated to be about 255, or about 20 de- 
 grees above the absolute zero.
 
 By taking advantage of the cooling of a gas produced by 
 its steady flow from an orifice without doing external work, 
 and by cooling the gas which is still under pressure by means 
 of the cooled gas, carried back past the pressure chamber, 
 Linde has succeeded in liquefying air in large quantities under 
 atmospheric pressure. The same method has also been used 
 for the liquefaction of other gases. 
 
 115. Relations of Meat and Mechanical L'ncryy. In what 
 we have studied up to this time we have ignored the question 
 of the nature of, heat. In describing the phenomena and in 
 stating the laws of the phenomena, it has not been necessary for 
 us to know more about heat than that it is something which 
 is associated with the sensation of temperature, and which 
 can be measured by the calorimeter. In this respect our 
 course has been exactly similar to that which was followed 
 in the actual development of the subject, for although the 
 conception of heat which prevailed during the time of that 
 development was given greater precision of statement than 
 we have found necessary, yet after all, it amounted to nothing 
 more than the conception we have used. During the century 
 in which most of the experimental work was done, heat was 
 commonly believed to be a substance, and hence indestructible, 
 which by entering bodies raised their temperatures. It was 
 also supposed to exist in separate and exceedingly minute 
 particles, which repelled each other, anu attracted the particles 
 of bodies. The additional hypothesis that it was not subject to 
 the force of gravity was rendered necessary by the fact that 
 the weight of a body does not depend on its temperature. The 
 substance thus conceived of was named caloric by Laplace and 
 Lavoisier. It is easy to see that by ascribing the right sort of 
 properties to such a substance, most of the phenomena w.hich 
 we have described can be explained. 
 
 A few facts, however, were known, of fundamental import- 
 ance for the theory, which coiild not be explained on this 
 hypothesis. Of these the most important and general one is 
 the production of heat by friction. Lord Bacon called atten- 
 tion to this, and made it one of the strongest points of the
 
 176 HKAT. 
 
 argument in which he contended that heat was not a sub- 
 stance, but the motion of the minute parts of bodies. 
 
 In 1798 Count Rumt'ord published an account of experi- 
 ments which he had made on the heat produced by friction. 
 He was at the time engaged in superintending the making of 
 cannon in the arsenal at Munich. His attention was attracted 
 to the great quantity of heat developed in the cannon when 
 it was being bored out, and especially by the high temperature 
 of the chips cut -out by the boring tool. The upholders of the 
 caloric theory of heat explained this high temperature by sup- 
 posing that the specific heat of the metal was diminished by 
 the friction upon it, so that the same quantity of heat in it 
 would cause a higher temperature. Rumford first showed 
 by direct measurement that the specific heat of the chips cut 
 off by the tool was the same as that of any block of the same 
 metal. He then mounted a cylinder of the metal in such a 
 way that it could be revolved around a blunted boring tool, 
 which pressed against the bottom of a deep cavity cut in it. 
 With this arrangement he found that as the block of metal 
 was turned, its temperature continually rose. The amount of 
 metal abraded by the blunt tool was so trifling that the rise 
 of temperature could not reasonably be ascribed to any 
 change in its specific heat, and Rumford concluded that it was 
 due to heat directly produced or made by the friction. jJy 
 surrounding the block of metal with water, he was able to 
 measure the quantity of heat thus produced. In Rumford's 
 mind the essential feature of the experiment was the contin- 
 ued production of heat in apparently unlimited quantities, 
 without any change occurring in the bodies producing it, by 
 which it could be accounted for. He stated his conclusion in 
 the following words: "It is hardly necessary to say that any- 
 thing which any insulated body, or system of bodies, can con- 
 tinue to furuish without limitation, cannot possibly be a 
 material substance; and it appears to me to be extremely 
 difficult, if not quite impossible, to form any distinct idea of 
 anything capable of being excited and communicated in the
 
 HEAT. 177 
 
 manner the heat was excited and communicated in these experi- 
 ments, except it be motion." 
 
 Almost at the same time Davy performed experiments of 
 a similar nature. He constructed a mechanism by which two 
 blocks of ice were rubbed together in a room whose tempera- 
 ture was below the melting point. He found that by this opera- 
 tion the ice was melted. He also constructed an arrangement by 
 which a brass disk was turned by clock-work against the fric- 
 tion of another piece of brass. Small pieces of wax were 
 placed on the disk, to show, by their melting, when the tem- 
 perature had risen to the melting point of the wax. This 
 apparatus he placed on a block of ice in the receiver of an air 
 pump, and exhausted the air from around it. He found that, 
 when the disk was turned for a while, the wax was melted, 
 and the arrangement of the experiment was such that the heat 
 which appeared could not have entered the disk from without. 
 He drew from these observations essentially the same conclu- 
 sion as that drawn by Rumford. 
 
 These experiments of Rumford and Davy were accepted by 
 some as proving that heat could not be a substance, and 
 therefore that, as Thomas Young said, "it must be a quality, 
 and this quality can only be motion." In general, however, 
 they had no effect in changing the prevailing theory f that heat 
 was a substance. The holders of that theory either ignored 
 them altogether, or set them aside with the expectation that 
 in the future they would be explained in a way consistent with 
 their theory. 
 
 In 1842 Robert Mayer published a paper in which he as- 
 serted the possibility of the transformation of mechanical 
 energy into heat, and the reversed transformation of heat into 
 mechanical energy. His argument was rather a metaphysical 
 than a physical one, and attracted no attention. He took, 
 however, one important step by attempting the calculation 
 of the amount of heat which is equivalent to, or which may 
 be transformed into, a unit of mechanical energy. 
 
 In the following year, Joule began an investigation of the 
 various ways in which heat may be produced by the expendi-
 
 ture of mechanical energy, in order to determine whether or 
 not the same amount of heat, in whatever particular way it 
 may be produced, is always produced by the expenditure of 
 the same amount of mechanical energy. The results obtained 
 by the different methods which he employed were not very 
 consistent with each other, though they were of the same 
 order of magnitude. They were sufficiently consistent, how- 
 ever, to convince Joule that heat may be produced by the 
 expenditure of mechanical energy, and that the amount of 
 heat produced is always in the same proportion to the amount 
 of energy expended. 
 
 The ratio of the energy expended to the amount of heat 
 produced by it, or the amount of energy which will produce 
 one unit of heat, is called the mechanical equivalent of heat. 
 Joule undertook the task of determining this quantity. To 
 do this he used a vessel filled with water, in which a paddle- 
 wheel could be revolved. Flanges projecting from the walls 
 of the vessel, between which the blades of the paddle passed, 
 broke up the circulation of the water, and greatly increased 
 the friction against which the paddle turned. The paddle was 
 kept in motion by a falling Ayeight. The mechanical energy 
 expended, or the work done by the falling weight, was deter- 
 mined from the known value of the weight and of the dis- 
 tance through which it moved, and the heat developed by the 
 friction of the paddle in the water was determined by an ob- 
 servation* of the rise of temperature. It was found that what- 
 ever work was done, the ratio of the work done to the heat 
 developed was constant. Joule used also a similar apparatus, 
 in which the vessel and paddle used were made of iron, and the 
 liquid used was mercury. He also used an apparatus in which 
 the friction which developed heat was that between two iron 
 plates rubbed together under mercury. From these three 
 forms of the experiment he obtained the same result, namely, 
 that the amount of work which will raise a pound of water 
 one degree Fahrenheit is 772 foot pounds. This statement is 
 equivalent to tTie following: That the amount of work which 
 will produce a calorie of heat is 423 kilogramme-metres. More
 
 recent determinations have increased these numbers some- 
 what. For ordinary work we may take 425 kilogramme- 
 metres as the mechanical equivalent of one calorie. 
 
 Joule's results carried conviction that heat is a form of 
 energy, and that it can be transformed into mechanical energy, 
 or mechanical energy transformed into it, without loss. 
 
 110. Conservation of Energy. Heat may be transformed 
 into work, or work into heat, by processes which are far more 
 complicated than the one which we have described. In certain 
 stages of these processes the condition of the system is often 
 such that we do not recognize in it the presence either of 
 mechanical energy or of heat. Whatever this condition may 
 be, it is found that a quantitative relation exists between it 
 and the work used, or the heat absorbed, to bring it about. 
 The first of Joule's experiments is a good example of one of 
 these processes. Joule began by investigating the relations of 
 heat to the electric current, and to the chemical action which 
 takes place in the voltaic cell, while the current is flowing. 
 The chemical effect in the cells which he used was the con- 
 sumption of zinc by the acid of the cell. He first determined 
 the amount of heat developed during the consumption of a 
 certain mass of zinc by direct chemical action, and compared 
 this with the amount of heat developed in an electric circuit, 
 when the same amount of zinc was consumed. He found that 
 the two quantities of heat were equal. The difference between 
 them was simply one of distribution, the heat in the first case 
 being developed directly at the place where the chemical action 
 was going on, while in the second case it was distributed 
 throughout the circuit. Joule next inserted in the circuit a 
 small magnetic motor. When the motor was at rest, or when 
 it was revolving without doing any work, the heat developed 
 in the circuit, for the consumption of an equal mass of zinc, 
 was the same as before. When, however, the motor was made 
 to lift a weight and so do work, less heat was developed in the 
 circuit, for the consumption of the same amount of zinc. The 
 amount of heat which apparently disappeared had evidently 
 been transformed into mechanical work. The process by which
 
 180 HBAT. 
 
 this transformation had taken place was not, however, so 
 direct a one as that in which heat is produced by friction. It 
 involved a series of intermediate conditions of the system, 
 which were neither mechanical nor thermal conditions. To 
 get a full survey of this operation, we should go back to the 
 fact that the zinc used in the cell was separated from an inert 
 chemical combination by the aid of heat. Beginning at that 
 point, the operation described was one in which the heat used 
 to obtain the pure zinc was transformed into mechanical 
 work. 
 
 To explain this series of conditions, and countless others 
 like them, we assume that energy exists in the universe in 
 several forms, and that the energy in any one form can be 
 transformed into any other. Thus, in the example before us, 
 the heat originally used in the furnace was turned into energy 
 of possible chemical combination, resident in the zinc and the 
 acid; this energy was transformed into electrical energy, and 
 this, in turn, either into heat or into mechanical work. So 
 far as we can judge by experiment, transformations of enei'gy 
 take place without any loss or gain of energy. 
 
 The general principle that there exists in the universe a 
 certain quantity of energy, which is unchanged by any nat- 
 ural operation, was first announced by Mayer. It was also 
 announced by Joule on independent grounds, and received its 
 first confirmation from his experimental work. The publica- 
 tion by Helmholtz, in 1847, of an important paper, in which 
 he discussed the principle in connection with its application 
 to various departments of physics, and showed that it was 
 consistent with all that was known about natural operations, 
 may be considered to mark the establishment of the principle. 
 
 In its general form, the principle is known as the conserva- 
 tion of energy. It may be stated as follows: In any closed 
 system, that is, in any system into which no energy enters and 
 out of which no energy goes, the amount of energy remains 
 constant, whatever be the transformations going on within 
 the system. To make this statement accord more nearly with 
 the conditions under which experiments must be carried on,
 
 HEAT. 181 
 
 we may say that the amount of energy in any system, what- 
 ever be the transformations which go on within the system, is 
 increased or diminished only by an amount equal to that of 
 the energy which passes through the bounding surface of the 
 system. A statement which we cannot contradict, although 
 it is one which we cannot prove, is that the energy of the 
 universe is a constant quantity. 
 
 117. Kinetic Theory of Heat. As an example of the use 
 of the inductive method of reasoning, Lor,d Bacon, in the 
 Novum Organum, considered the nature of heat. On grounds 
 which would now be considered very unsatisfactory, he came 
 to the conclusion that heat was a peculiar kind of motion of 
 the small particles of bodies. Xewton, and his contemporary, 
 Locke, held the same view. The theory that heat is an im- 
 ponderable substance was, however, so easy to work with, and 
 satisfied so fully most of the demands made upon it by the 
 discoveries of the time, that it displaced the other theory, ex- 
 cept in the minds of a few leading thinkers, like Cavendish and 
 Thomas Young. We have seen how Rumford and Davy were 
 led to adopt a view similar to Bacon's by reflecting upon the 
 results of their experiments. Joule's thought on the matter 
 took the same course. When it was proved by Clausius and 
 Maxwell that the properties of gases could be explained by 
 the kinetic theory, it became almost a matter of course to ex- 
 tend this theory to all the states of matter, and to account for 
 their relations to heat by the motions of their molecules. 
 
 The method employed by Mohr may be used to confirm our 
 belief in the general kinetic theory of heat. It consists in 
 assuming that the molecules of all bodies are in motion, and 
 that this motion is increased when heat enters the body, and 
 in tracing the consequences of this assumption to see whether 
 it will afford reasonable explanations of the properties of 
 bodies with respect to heat. For example, we may explain 
 the expansion of bodies by heat, by ascribing it to their in- 
 creased molecular motions, by means of which each molecule 
 is able to force its neighbors away from itself a little farther. 
 We explain conduction of heat as the transfer of kinetic energy
 
 182 UKAT. 
 
 by impact from the hotter or more rapidly moving molecules 
 to the colder ones near them. We explain evaporation in the 
 way which has already been given in 113. We explain the 
 cooling of a gas when it expands against external pressure, by 
 supposing that the work that is done by that expansion is de- 
 rived from the kinetic energy of its molecules. We explain 
 the sensation of heat by supposing it due to the impacts of 
 the molecules against the ends of the nerves. In general, when 
 mechanical work is transformed into heat, we suppose it to be 
 transformed into the kinetic energy of the molecules of the 
 heated body, and into the potential energy which those mole- 
 cules acquire by reason of the expansion of the heated body. 
 There are some operations, as, for example, the melting of a 
 solid, which cannot be explained completely without additional 
 assumptions. These assumptions, however, are not incon- 
 sistent with the kinetic theory, but extend it by considering 
 the motion of the atoms as well as that of the molecules. 
 
 So long as we consider only heat and mechanical energy, 
 we may obtain a consistent account of their relations by con- 
 sidering kinetic energy to be the only real energy. For if we 
 assume that there exists another medium or form of matter, 
 which our senses cannot perceive, and that this medium is in 
 motion, we may explain all forms of the potential energy of 
 bodies by supposing them due to the kinetic energy of a larger 
 system, consisting of those bodies and of this hypothetical 
 medium. The tendency in physical speculation has been, for 
 some time, to attempt to explain all physical phenomena by 
 ascribing them to the motions either of tangible bodies or of 
 this hypothetical intangible medium. That is, the attempt has 
 been made to construct what may be called a mechanical model 
 of the physical universe. Recent researches, especially those 
 connected with the electrical discharge and the behavior of 
 radioactive bodies, have indicated the possibility of explaining 
 the energy of matter by ascribing it to the motions in a 
 medium of separate portions of electricity. This explanation 
 has not yet been worked out, but it is at least a possible one,
 
 HEAT. 183 
 
 and there is little doubt that for some time speculative physics 
 will work along the new line indicated by it. 
 
 118. Laics of Thermodynamics. When heat is transformed 
 into work, the transformation is effected by means of a body 
 which, in the discussion that follows, we shall call the working 
 body. In order to study the transformation with as little 
 complication as possible, we consider it effected by a series of 
 changes which bring the physical condition of the working 
 body and of the other parts of the system back again to that 
 in which they were before the operation began. Such an 
 operation is called a cyclic operation, or a cycle. At the end 
 of a cycle the energy of the working body is the same as it 
 was at the beginning, so that the heat which the working body 
 has received has been either given out by it, or transformed 
 into work. The use of such cycles was introduced by Sadi 
 Carnot, in 1824. The simple form of cycle, also proposed by 
 Carnot, we shall call a Carnot cycle. In performing it the 
 working body is used in connection with two limitless bodies, 
 at different temperatures, with which the working body can 
 exchange heat. The hotter of these two bodies is called the 
 source, the cooler, the refrigerator. This set of bodies is called 
 a thermodynamic engine. To put the engine through a Carnot 
 cycle, we take the working body at the temperature of the 
 refrigerator, and compress it while it is enclosed in an envelope 
 impermeable to heat. Its volume will diminish, and its tem- 
 perature will rise, while mechanical work is done upon it from 
 without. This first operation is' called an adiabatic com- 
 pression. Compression is continued until the temperature of 
 the working body rises to that of the source. The working 
 body is then put in contact with the source, and allowed to 
 expand. As it does so, heat flows in from the source, so as to 
 keep the temperature of the working body constant. At the 
 same time the working body does work by expanding against 
 external pressure. This operation is called an isothermal ex- 
 pansion. It is allowed to continue until any desired quantity 
 of heat is withdrawn from the source. The working body is 
 then enclosed in an envelope impermeable to heat, and Is
 
 184 HEAT. 
 
 allowed to expand still further. During this adiabatic ex- 
 pansion, it does additional work against external pressure and 
 its temperature falls. The expansion is allowed to continue 
 until the temperature of the working body becomes that of 
 the refrigerator. The working body is then put in contact 
 with the refrigerator, and is compressed. Heat flows from it 
 into the refrigerator, so as to keep its temperature equal to 
 that of the refrigerator. Work is done upon it during this 
 operation. This isothermal compression is continued until 
 the volume of the working body becomes again that which it 
 was at the beginning of the operations. The working body i^ 
 thus brought to the same condition as that in which it was 
 at first, and the series of operations which it has performed 
 constitutes a cycle. 
 
 In performing the second and third operations of the cycle, 
 the working body has done external work; in performing the 
 fourth and first operations, it has had work done on it. The 
 whole amount of work done during the cycle is the difference 
 between these two quantities, and is positive ; that is, the work- 
 ing body has performed external work. 
 
 In the study of thermodynamics, the principle of the 
 equivalence of heat and energy is called the first law of ther- 
 modynamics. It may be stated as follows: When heat is 
 transformed into work, or work into heat, the quantities of 
 heat and work, when measured in the proper units, are equal 
 to each other. When we apply this law to the cyclic opera- 
 tion just described, it is evident that the work done during 
 the cycle must have been due to the transformation of heat 
 in the working body, and since the heat in the working body 
 is the same at the end of the cycle as it was at the beginning, 
 the heat transformed must have been part of that received 
 from the source. The work done is therefore equal to the 
 difference between the heat which leaves the source and the 
 heat which enters the refrigerator. 
 
 The cyclic operation which has been described is reversible. 
 That is, the series of operations can be described in the reverse 
 order, and so that every feature of the cycle is exactly re-
 
 HKAT. 185 
 
 versed. The fourth operation, for example, becomes on re- 
 versal an isothermal expansion, by which the same amount of 
 heat is received by the working body at the temperature of the 
 refrigerator as was sent into the refrigerator during the 
 isothermal compression, and the work done by the working 
 body during this expansion is equal to the work done on it 
 during the former compression. Similarly it may be shown 
 that the other steps of the cycle are also reversible. 
 
 Clausius and Lord Kelvin proved a most important theorem 
 regarding the operation of a reversible cycle. Carnot first 
 asserted it, but his demonstration of it was erroneous. It is 
 generally known as Carnot's theorem. We may state it as 
 follows : The efficiency of any reversible engine is greater than 
 that of any other engine working between the same source and 
 refrigerator. By efficiency is meant the ratio of the work done 
 by the engine to the heat received from the source. The proof 
 of this theorem depends upon a general principle, which we 
 shall call the second law of thermodynamics. To lead up to 
 this principle, let us consider a cyclic operation performed by 
 two engines, one of which is reversible, while the other is not. 
 We assume, for the sake of argument, that the non-reversible 
 engine is more efficient than the other one, and we gear the 
 engines together so that the non-reversible engine, working 
 forward, spends all its work in driving the reversible engine 
 backward. On the whole, then, no work will be done by the 
 two engines. We have supposed the engine which works for- 
 ward to be the more efficient. It will therefore have to take 
 from the source less heat to do a certain amount of work than 
 the reversible engine will, and therefore when that work is 
 spent in driving the reversible engine backward, the reversible 
 engine gives up to the source more heat than the non-reversible 
 engine takes from it. Since the first law applies to both 
 engines, it follows that the non-reversible engine delivers less 
 heat to the refrigerator than the reversible one takes from it 
 when it is run backward. The final result of the performance 
 of such a cycle as this, which can be repeated as many times 
 as we please, is to transfer heat from the refrigerator to the
 
 186 HEAT. 
 
 source, while at the same time no work is done, and no change 
 occurs in the internal condition of the bodies which constitute 
 the system. The result thus reached would require us to be- 
 lieve that heat would pass of itself, or without any comper- 
 sating change in the physical condition of the system, from 
 the colder parts of the system to its hotter parts. Now, in 
 our experience, heat does not behave in this way. It passes 
 always from hotter bodies to colder bodies, and not in the 
 reverse sense. 
 
 Clausius embodied the general belief on this subject in the 
 following statement, called the second law of thermodynamics- 
 Heat cannot pass of itself, or without compensation, from a 
 colder to a hotter body. 
 
 If we accept this law as valid, we deny the possibility of 
 the result obtained by the use of the two engines whose 
 operation we have described, and we accordingly deny the 
 validity of the hypothesis upon which that operation depended, 
 namely, the hypothesis that the efficiency of the non-reversible 
 engine was greater than that of the reversible engine. The 
 efficiency of the reversible engine is therefore the greatest 
 possible. 
 
 It is easy to extend this demonstration so as to show that 
 the efficiency of all reversible engines, which work between the 
 same source and refrigerator, is the same. The efficiency of 
 the reversible engine is therefore independent of the particular 
 material used in the working body. It depends only on the 
 temperatures of the source and the refrigerator. In other 
 words the efficiency of the Carnot's engine is a function of the 
 temperatures of the source and refrigerator. 
 
 119. Absolute Scale of Temperature. In the definition of 
 the scales of temperature which we have hitherto used, we 
 avail ourselves of some property of a body which is a function 
 of its temperature. For example, in our definition of the 
 Centigrade scale, we make use of the volume of mercury con- 
 tained in glass. In this instance, as in all the others, the prop- 
 erty employed is that of a particular body, or at best, of a 
 particular substance.
 
 HEAT. 1ST 
 
 We have now discovered, by the study of the Carnot cycle, 
 a property, depending only on temperature, which is not pecu- 
 liar to any one body or to any one substance, but is common 
 to all. We may therefore use this property as a means of 
 denning a scale of temperature, in such a way as to make our 
 definition independent of any particular substance. The scale 
 which we shall thus define is called the absolute scale of tem- 
 perature. 
 
 A complete definition of the absolute scale is obtained by 
 the choice, first, of the form of the temperature function 
 which measures the efficiency, and secondly, by a determina- 
 tion of the length of the degree, or of the numerical value of 
 some standard temperature, on that scale. 
 
 One general remark may here be made, which holds true, 
 whatever be the form of the temperature function chosen. 
 The efficiency of an engine working from any given source will 
 increase as the temperature of the refrigerator falls; but it 
 can never exceed unity, for it is manifestly impossible for 
 more work to be done by the engine than is equivalent to the 
 heat received by it from the source. The temperature of the 
 refrigerator for which the efficiency becomes unity is there- 
 fore the lowest conceivable temperature. There can be no 
 lower temperature than that, and it must be the same in all 
 bodies which will serve, when used as refrigerators, to make 
 the efficiency equal to unity. This temperature is therefore a 
 natural zero, and it is advisable, in constructing the absolute 
 scale, to mark it as the zero point of that scale. All actual 
 temperatures will then be positive. 
 
 Turning now to the choice of the particular temperature 
 function by which the efficiency shall be expressed, we are 
 guided in making it by the study of the efficiency of an engine 
 in which the ideal gas is used as the working body. If we 
 express the temperatures of the source and refrigerator on 
 the scale of the ideal gas thermometer, we find that the effi- 
 ciency of such an enfine is equal to the difference between 
 the temperatures of the source and of the refrigerator, divided 
 by the temperature of the source. That is, if we represent
 
 188 HEAT. 
 
 these temperatures by 8 and R respectively, the efficiency is 
 
 expressed in terms of them by the formula, S R . Since 
 
 S 
 
 the efficiency does not depend on the nature of the work- 
 ing body, this expression is also the measure of the efficiency 
 when any other substance than the ideal gas is used as the 
 working body. This relation so commends itself by its sim- 
 plicity, that we adopt it as the general relation, or form of 
 the temperature function, by which the efficiency is ex- 
 pressed. That is, we suppose the efficiency to be known, and 
 define the temperatures of the source and refrigerator by a 
 formula similar in form to that just given. 
 
 From this formula it appears that the efficiency of the 
 engine becomes unity when the temperature of the refrigerator 
 becomes the zero of the ideal gas thermometer. This zero is 
 therefore the lowest conceivable temperature, and we may 
 adopt it as the absolute zero, or zero of the absolute scale. 
 
 It remains to adopt the length of a degree on the absolute 
 scale. It is convenient to have this length as nearly as pos- 
 sible the same as the length of the Centigrade degree. We 
 accordingly adopt 100 degrees as the difference of absolute 
 temperature between the melting point and the boiling point. 
 By experiment we know that the efficiency of an engine work- 
 in A 
 
 ing between these temperatures is _!_ and hence, using the 
 
 373 
 
 general formula for efficiency and the convention just made, 
 we obtain 373 for the absolute temperature of boiling water 
 and 273 for the absolute temperature of melting ice. After 
 the numerical value of one temperature is fixed, the numerical 
 value of any other temperature is found by determining the 
 efficiency of an engine working between the two temperatures, 
 and using the general formula. The expansion of mercury in 
 glass is so nearly proportional to the change of absolute tem- 
 perature, that we may obtain a close approximation to the 
 absolute temperature of a body by adding 273 to its tempera- 
 ture on the Centigrade scale.
 
 HEAT. 189 
 
 We have thus obtained a complete scale of temperature, 
 starting from the absolute zero, and determined in a way 
 which is independent of the properties of any particular sub- 
 stance. It is the same as the scale of the ideal gas thermom- 
 eter, but has not the hypothetical character of that scale. 
 The kinetic theory of gases indicates that absolute tempera- 
 ture is a measure of the kinetic energy of the molecules of a 
 gas, and perhaps of other bodies. On this view the absolute 
 zero is the temperature of a body whose molecules are at rest. 
 
 120. Thcrmodynamic Properties of Bodies. When the 
 properties of bodies are examined by the methods of thermo- 
 dynamics, many important relations are found among them, 
 which are confirmed by experiment. For example, it may be 
 proved, as a general principle, that a substance, which ex- 
 pands under constant pressure when its temperature rises, 
 has its temperature raised by an adiabatic compression; and 
 that a substance, which contracts when its temperature rises, 
 has its temperature lowered by adiabatic compression. This 
 general conclusion was confirmed by Joule for the case of 
 water. In the intorval between <> und 4 Ontiirniclp. within 
 which water contracts as its temperature rises, a sudden com- 
 pression will lower its temperature. At any temperature 
 higher than 4, a sudden compression will raise its temperature. 
 
 As another example, we may take the way in which the 
 melting point of ice depends upon pressure. It may be proved, 
 as a general principle, that the melting temperature of a sub- 
 stance, that is, the temperature at which the solid and liquid 
 states nf the substance are in equilibrium, will rise if the 
 pressure is increased, provided the relative density of the 
 solid is greater than that of the liquid; but it will fall if the 
 pressure is increased, provided the relative density of the solid 
 is less than that of the liquid. This principle was confirmed 
 for the case of water by the experiments of James Thomson 
 and his brother, Lord Kelvin. Water is a substance belonging 
 to the second class, for which the density of the solid is less 
 than that of the liquid. When a mixture of water and ice, 
 whose temperature under atmospheric pressure is Centi-
 
 1!)0 HEAT. 
 
 grade, was subjected to additional pressure, its temperature 
 fell. The temperature rose again when the pressure was re- 
 moved. By using this principle we explain the melting of ice 
 under pressure, and its return to the solid state, or its rege- 
 lation, when the pressure is removed. Other substances, for 
 example, paraffine, which belong to the first class, were shown 
 by Bunsen and by other observers to exhibit a rise of temper- 
 ature when pressure was applied to them. 
 
 The methods of thermodynamics have been applied, not 
 only to the study of the general properties of bodies, but more 
 particularly to the study of solutions, and of mixtures of 
 different substances which act on each other chemically. In 
 this way a foundation has been laid for the physical study of 
 chemical action. It should be said that the results which have 
 been obtained do not furnish an explanation of chemical 
 action. They merely enable us to classify chemical actions 
 under certain general statements, which depend upon the 
 validity of the second law of thermodynamics. An explanation 
 of those actions would require an explanation of that law. 
 
 121. Dissipation of Energy. According to tlit- general 
 principle of conservation, the energy of the universe is con- 
 stant. Not all of it, however, is available for doing work. In 
 all cases of the transformation of energy in which non-con- 
 servative forces act, some at least of the energy appears in 
 the form of heat. If such transformations were continued 
 long enough, all other energy in the universe would be trans- 
 formed into heat, and if this heat could be transformed into 
 mechanical energy, all the energy of the universe would be 
 available for use as mechanical energy. But this is not the 
 case. We know, from our study of the Carnot engine, that 
 even when heat is transformed into work in the most efficient 
 way possible, some of the heat is not transformed, but is trans- 
 ferred from a higher to a lower temperature. None of the 
 heat in the body at lower temperature can be utilized, unless 
 a still colder body can be found to serve as refrigerator. It 
 is plain, therefore, that the final effect of all transformation's 
 of energy will be to bring about a common temperature of ail
 
 HEAT. 191 
 
 bodies. When this common temperature has been attained, no 
 further use can be made of the heat in those bodies, and since, 
 by hypothesis, all other available energy has been turned into 
 heat, no available energy will be left. The amount of energy 
 in the universe will be still the same, but none of it can be 
 used. The process by which this final condition will be at- 
 tained is called the dissipation of energy. . 
 
 Boltzmann has explained the second law of thermodynamics 
 by the aid of the kinetic theory of heat. On the assumption 
 that the atoms of all bodies are in unordered motion, he has 
 shown that the most probable condition of any assemblage of 
 atoms is one in which the mean kinetic energy of each atom in 
 the same, and that, when this condition does not exist, the 
 probable change in the assemblage is toward this condition, 
 and not away from it. He considers that the second law of 
 thermodynamics is the experimental equivalent of this theo- 
 retical conclusion, and expresses the general tendency of bodies 
 toward this most probable condition. Boltzmann suggests a 
 possible escape from the conclusion that the final condition of 
 the universe is that in which there is no available energy, by 
 calling attention to the fact that, although the general ten- 
 dency of any system is toward its most probable condition, vet 
 there may occur in the system such a combination of condi- 
 tions that for a while the tendency will be toward an improb- 
 able condition. During this period the second law will not 
 hold true, and at the end of it, the system will have acquired 
 a store of available 7 energy.
 
 192 
 
 LIGHT. 
 
 122. General Considerations Respecting Light. The sense 
 of sight presents to us a large number of phenomena which 
 we associate with the general idea of light. Our object is to 
 study and classify these phenomena, and so far as possible to 
 explain them. We shall find in this case, for the first time, 
 that a sufficient explanation cannot be given in terms of the 
 motion of matter, and that it will be necessary to assume an 
 additional constituent of the physical universe as the medium 
 in which the actions, take place which constitute light. We 
 shall find further that this medium has properties unlike 
 those of any known body and that the actions in it do not 
 conform to the laws of action of any known body. Our ex- 
 planation therefore consists finally in the reduction of all the 
 phenomena of light to actions which take place in a hypo- 
 thetical- medium according to certain formal laws. 
 
 We shall find it expedient, in our study of this subject, to 
 examine by experiment most of the fundamental and im- 
 portant facts connected with it, before we undertake to de- 
 velop any theory of light. 
 
 123. Origin of Light. Opinions were divided, in antiquity, 
 between the view that light was an affection of external 
 bodies, passing from them to the eye, and the view that the 
 eye itself was the source of light, which passed out from it 
 to any object at which the eye was looking. The latter view 
 was unable to meet the criticism of Aristotle, to the effect 
 that, if it were true, we should be able to see in the dark, 
 except by artificial additions to the theory. It nevertheless 
 held a prominent place in thinking for many years after that 
 criticism was made on it. Gradually, however, the other view 
 displaced it. 
 
 The eai'liest observations must have shown that light 
 travels in straight lines.. This fact is so well known that we 
 make use of it in countless ways, without considering its im- 
 portance to the theory of light. In our subsequent discussions
 
 LIGHT. 193 
 
 we shall use the word ray to denote any straight line in which 
 light travels. 
 
 Another observation, which was made so early that no 
 record of it has come down to us, is that of the reflection of 
 light. When light, in passing from an object, encounters a 
 smooth surface, it is more or less completely reflected from 
 that surface. If we select a point on the reflecting surface 
 and draw from it a perpendicular to that surface, and if we 
 then draw from the source of light the ray to that point, this 
 ray will make with the normal an angle, which we call the 
 angle of incidence. The light from the source is reflected at 
 the point of incidence according to the following law: The 
 incident and reflected rays lie in a plane which contains the 
 normal to the reflecting surface, and the angles of incidence 
 and reflection are equal. 
 
 A third observation, which for the present we shall de- 
 scribe only in general terms, is that of the refraction of light 
 This refraction is the sudden bending or change of direction 
 which a ray of light exhibits as it passes from one medium 
 into another. It is refraction which makes a straight stick 
 look bent when one end of it is thrust into water. 
 
 124. Mirrors. A mirror is, in general, a polished surface 
 used for the reflection of light. The surface usually conforms 
 very nearly to some geometrical surface. The most common 
 mirrors have either plane or spherical surfaces. 
 
 In the case of the plane mirror, if we consider a point- 
 source of light set up in front of it, and trace the course of 
 the rays, which are reflected from various parts of it, by 
 means of the general law of reflection, we shall find that they 
 all diverge from a point which is situated behind the mirror 
 on the perpendicular drawn from the source to the mirror, 
 and as far behind the mirror as the source is in front of it. 
 This point is called the image of the source. 
 
 The eye sees an object on the line along which the ray of 
 light from that object enters the eye. If we consider an ex- 
 tended object placed before a plane mirror, and trace the 
 course of the rays, from the different points of the object,
 
 194 LIGHT. 
 
 which reach the eye after reflection from the mirror, we shall 
 find that they proceed from images of those points, situated 
 behind the mirror, according to the rule just given; and ac- 
 cordingly, that the eye perceives in the mirror a collection of 
 images, presenting an exact counterpart of the object, as it 
 would appear if it were transferred to the place of its image, 
 with the exception that the image is reversed, the right hand 
 of the object appearing to be the left hand of the image, and 
 vice versa. 
 
 To study the spherical mirror, we suppose a small portion 
 of a spherical shell to be cut off by a plane section. The 
 diameter of the sphere prolonged through the middle point of 
 the section we call the axis. If reflection occurs at the inner 
 surface of this spherical cap, it is a concave mirror ; if at the 
 outer surface, a convex mirror. The point where the axis cuts 
 the mirror is called the vertex. The centre of the sphere, of 
 which the mirror forms a part, is the centre of the mirror, 
 and the radius of that sphere is the radius of the mirror. 
 
 We shall consider first the case of the concave mirror. If 
 we suppose the source of light to be a point on the axis more 
 distant from the mirror than its centre, and if we trace the 
 incident and reflected rays from a point on the mirror, which 
 make equal angles with the radius drawn to that point, we 
 find, of course, that the reflected rays cut the axis at a point 
 lying between the vertex and the centre. We suppose the 
 mirror to be small, so that the angle between the incident 
 ray and the axis is always small. With that limitation it 
 may be proved that whatever be the point on the mirror at 
 which the incident ray meets it, the reflected ray will always 
 pass through the same point on the axis. To do this we in- 
 vestigate the distance from the vertex of th<* point on the 
 axis at which the reflected ray intersects it. Let us represent 
 the distance of the source from the centre by a, the distance of 
 the point of intersection from the centre by 6. Let us repre- 
 sent further the radius of the mirror by r, and the distances 
 from a point on the mirror to the source and the point of inter- 
 section respectively, by f and /'. Because the angles of inci-
 
 LIGHT. 195 
 
 dence and reflection are equal, we have among the sides of the 
 triangles the proportion f:f'=a:b. Now, if the mirror is 
 small, f and f are equal, to a first approximation, to the dis- 
 tances from the vertex to the source and the point of intersec- 
 tion respectively. We have therefore, a=f r, and b=r f, 
 and using these values in the proportion already given, we 
 obtain fr-ff'=ff'f'r. Prom this we obtain (/ + f')r - 2ff', 
 and finally \+j,= \- 
 
 This equation gives the distance of the point of intersection 
 from the vertex in terms of the distance of the source from 
 the vertex and of the radius of the mirror; that is, in terms of 
 quantities which are independent of the angle between the inci- 
 dent ray and the axis. The same point, therefore, will be a 
 common point of intersection of all reflected rays coming from 
 the given source. It is called the focus of the source. 
 
 It is plain from the construction that the source and focus 
 may be interchanged. The two points thus related are called 
 conjugate foci. 
 
 If the source is at an infinite distance from the mirror, the 
 formula shows that the focus will be midway between the 
 centre and the vertex. The point thus determined is called 
 the principal focus of the mirror. As the source moves up 
 toward the mirror from infinity, the focus moves toward the 
 centre. When the source reaches the centre, it coincides with 
 the focus. After it has passed the centre, the focus moves out 
 toward infinity, and is at an infinite distance when the source 
 is at the principal focus. During all these changes the focus 
 is a point through which the reflected rays actually pass. It 
 is therefore called a real focus. 
 
 If the source is between the principal focus and the vertex, 
 the formula shows that the distance of the focus from the 
 vertex is negative. This indicates that the focus stands be- 
 hind the mirror. It may be shown by construction that, in 
 this case, the reflected rays diverge from the mirror. They 
 appear to come from a point behind it. This point, through 
 which the rays do not actually pass, is called a virtual focus.
 
 196 LIGHT. 
 
 If the mirror is convex instead of concave, a construction 
 similar to the one already made will show that the reflected 
 rays always diverge from the mirror, and that the focus of 
 any point on the axis is a virtual focus. A demonstration 
 similar to the one already given shows that a formula holds 
 which is similar to the one already obtained, except that the 
 terms containing lines drawn to the other side of the mirror 
 have the minus sign. A study of the formula shows that the 
 principal focus lies midway between the centre and the vertex, 
 and that the focus of any real point on the axis lies between 
 the principal focus and the vertex. 
 
 125. Images. If a small extended object stands transverse 
 to the axis, a line may be drawn from each point of it through 
 the centre to the mirror, which will be an axis for that point, 
 and the focus of the point will lie on that axis, or on its pro- 
 longation through the mirror. The assemblage of foci found 
 in this way for the different points of the object constitutes 
 what is called an image of the object. According as the foci 
 which form it are real or virtual, the image is called a real or 
 virtual image. Construction shows, in the case of the concave 
 mirror, that, so long as the object lies outside the principal 
 focus, the image is real and inverted. When the object lies 
 within the principal focus, the image is virtual and erect. In 
 the case of the convex mirror, the image is always virtual and 
 erect. 
 
 Real images can always be studied by throwing them upon 
 screens. Virtual images can only be examined by the eye. 
 
 126. Mirrors of Large Aperture. The solid angle subtended 
 by a spherical mirror from its centre is called its aperture. 
 The mirrors which we have considered hitherto, and to which 
 our former statements apply, were of very small aperture. 
 When the aperture becomes considerable, the approximations 
 upon which those statements depend no longer hold good. 
 We find in such cases that no definite focus exists. The rays 
 reflected from the different points of the mirror cut each other 
 in such a way as to form a specially illuminated surface. In 
 the plane diagram this becomes a special line called the 
 caustic.
 
 LIGHT. 197 
 
 When the source is not placed on the axis the reflected 
 light passes from a small portion of the mirror in such a way 
 as to determine a line in space through which the rays pass, 
 and which is in consequence specially illuminated. This is 
 called the first focal line. It is at right angles to the plane 
 of incidence. After passing through this they again determine 
 a specially illuminated line, which is in the plane of incidence. 
 This is called the second focal line. 
 
 127. Refraction. Attention has already been called to the 
 fact that when a ray of light passes from one medium into 
 another, it is refracted or bent at the surface which separates 
 the media. The ray in each medium is straight. For very 
 many years the attempts which were made to discover any law 
 governing this refraction were fruitless. The Greek astronomer 
 Ptolemy was the first to investigate this question, by the study 
 of the refraction between air and water, air and glass, and 
 water and glass. He could find no general law, and was forced 
 to content himself with an empirical table, giving the angles 
 of refraction corresponding to certain angles of incidence. 
 Even Kepler, who investigated the same question by the help 
 of the measurements of Ptolemy and others, did not perceive 
 the true law. It was first discovered by Snell and was first 
 published by Descartes. It may be stated as follows: The 
 incident and refracted rays lie in a plane which contains the 
 normal to the refracting surface, and the ratio of the sine of 
 the angle of incidence to the sine of the angle of refraction is 
 constant. The numerical value of this constant depends upon 
 the nature of the media which lie on either side of the refract- 
 ing surface. When the media are specified, both as to their 
 nature and as to the order in which they are considered, the 
 constant is called the index of refraction from the first to the 
 second. Thus, if we consider the incident ray in water, re- 
 fracted at a surface where the water meets a block of glass, 
 we call the ratio of the sines of the angles of incidence and 
 refraction in this case the index of refraction from water to 
 glass. If only one medium is specified, it is assumed that it is 
 the second medium, in which is the refracted ray, and it is
 
 198 LIGHT. 
 
 assumed that the first medium is either air, or better, vacuum. 
 The constant in this case is called the index of refraction of 
 the second medium. For example, when light passes from air 
 into water, the ratio of the sines is 1.333. This number is 
 called the index of refraction of water. 
 
 Experiment shows that, for all substances with which we 
 are at present concerned, the index of refraction is greater 
 than unity, so that the angle of refraction is less than the 
 angle of incidence. If the index of refraction between two 
 media is greater than unity, the second medium, in which is 
 the refracted ray, is said to be optically denser than the other. 
 The term is a convenient one, if we are careful not to take it 
 to mean the actual density of the medium. 
 
 If the refracted ray is turned back on itself by reflection 
 in a plane mirror, or if a source of light is placed in the line 
 of the refracted ray, the ray which proceeds through the 
 second medium will be refracted at the original point of inci- 
 dence, and the refracted ray thus obtained will coincide with 
 the original incident ray. The index of refraction between 
 two media when they are taken in one order, is therefore the 
 reciprocal of the index of refraction when they are taken in 
 the reverse order. 
 
 Let us consider for a moment the simple case of refraction 
 between air and water, which is a typical one. When the inci- 
 dence is perpendicular, the direction of the refracted ray is 
 the same as that of the incident ray. As the angle of inci- 
 dence, in the air, increases, the angle of refraction, in the 
 water, increases also, though not so rapidly. The ratio of the 
 sines of the two angles is always the same number. When the 
 angle of incidence becomes a right angle, which is as large as 
 it can be, the angle of refraction is less than a right angle, 
 and the sine 'of the angle of refraction equals the reciprocal 
 of the index of refraction. Consider next the refraction of a 
 ray which is incident in water on the surface separating the 
 water from the air. When the incident ray is perpendicular 
 to that surface, the direction of the refracted ray is un- 
 changed. As the angle of incidence, in the water, increases,
 
 the angle of refraction, in the ah-, also increases, but more 
 rapidly. The ratio of the sines of the two angles is the re- 
 ciprocal of the index of refraction of water. When the angle 
 of incidence is so great that its sine is equal to the reciprocal 
 of the index of refraction, the sine of the angle of refraction 
 is equal to unity, and the angle of refraction is a right angle. 
 The refracted ray, in this case, just emerges from the water 
 into the air. If the angle of incidence is made still greater, it 
 follows from the formula that the sine of the angle of re- 
 fraction is greater than unity. This impossible result indi- 
 cates a failure of the law of refraction. \n fact, after the 
 angle of incidence has passed the limiting value, for which its 
 sine is equal to the reciprocal of the index of refraction, light 
 no longer emerges into the air. This limiting value of the 
 angle of incidence is called the critical angle. Light incident at 
 an angle which is greater than the critical angle is totally 
 reflected within the water. 
 
 128. Fermat's Law of Least Time. A general law was 
 announced by Fermat, which governs both reflection and re- 
 fraction. It may be stated by saying, that the time taken by 
 light to pass from one point to another by way of a reflecting 
 or refracting surface is a minimum. To illustrate this for the 
 case of reflection, let us consider two points on the same side 
 of a reflecting surface, from one of which the incident ray 
 starts. The other point is the one through which the reflected 
 ray passes. If we select any point on the reflecting surface, 
 and draw lines from it to the two points already chosen, it 
 may be shown that the sum of the lengths of these lines is a 
 minimum when the point on the surface is so placed that the 
 lines drawn to it from the other points lie in a plane contain- 
 ing the normal to the surface and make equal angles with that 
 normal. The lines whose lengths fulfill the condition that 
 their sum is a minimum thus conform to the law of reflection, 
 and represent the incident and reflected rays which pass 
 through the two points. If light travels in a particular 
 medium with a definite velocity, the time required for it to 
 pass from the one point to the other by the lines which fulfill 
 the minimum conditions is the least possible.
 
 '200 LIGHT. 
 
 In the case of refraction Fermat assumed that the rate at 
 which light travels is different in different media, and that 
 the ratio of the velocities in any two media is equal to the 
 index of refraction between those media. If we select two 
 points, one to serve as the source from which the incident 
 ray comes, the other to serve as the point through which the 
 refracted ray passes, it may be shown that this law of the 
 velocity leads to the conclusion that the path along which 
 light will pass from the one point to the other in the least 
 time is that which conforms to the law of refraction. 
 
 This general principle is known as the principle of least 
 time. As announced by Fermat it was simply an hypothesis, 
 for which no experimental proof or even theoretical argument 
 could be given. It was subsequently shown to be a conse- 
 quence of the wave theory of light, and may now be used with 
 confidence in the solution of problems. 
 
 129. Prisms. A block of any transparent substance, en- 
 closed between two planes which meet at an edge, is called a 
 prism. The angle between the planes is called the refracting 
 angle of the prism. We shall consider only the simple case in 
 which the substance of the prism is glass, and the surrounding 
 medium air. If a ray of light is incident obliquely upon one 
 face of this prism, it is refracted in the glass toward the nor- 
 mal to the first surface, and travels on in the glass until it 
 meets the second surface. There it is again refracted, this time 
 away from the normal to the second surface. The result of 
 these two refractions, at least for many cases of incidence, is 
 that the two refractions combine to make the emerging refract- 
 ed ray deviate from the original direction of the incident ray, 
 toward the base of the prism. Of course, if the original inci- 
 dence is such that the refracted ray in the glass meets the 
 second surface so that its angle of incidence there is greater 
 than the critical angle, the ray will not emerge from the prism. 
 Analysis shows that the deviation of the emergent ray from 
 the direction of the incident ray is least when the refracted 
 ray in the glass is at right angles to the line which bisects 
 the refracting angle of the prism. When the prism is so placed
 
 LIOHT. 201 
 
 that this condition obtains, it is said to be in the position of 
 minimum deviation. As a prism is usually a block whose 
 cross-section is a triangle, with its base perpendicular to the 
 line bisecting the refracting angle, the condition of minimum 
 deviation is often described by saying that in it the refracted 
 ray in the glass is parallel with the base of the prism. 
 
 130. Refraction at a Single Spherical Surface. If light is 
 incident from a point in air upon a small portion of a spherical 
 surface bounding another medium, so placed that one of the 
 rays from the source meets that surface perpendicularly, it 
 may be shown that all the refracted rays will pass through a 
 point on that perpendicular line or on it produced into the 
 second medium. This point is a focus. If the focus is in the 
 second medium, the refracted rays actually pass through it, 
 and it is a real focus. If it is in the air, the refracted rays do 
 not pass through it, but appear to diverge from it, and it is a 
 virtual focus. The position of the focus depends not only on 
 the position of the source, and on the radius of the spherical 
 surface, but also on the index of refraction of the medium. 
 
 As a typical case we consider the spherical surface concave 
 toward the source. In this case we may show by construction 
 that the focus lies in the air, or is a virtual focus. The form- 
 ula connecting the distance f of the focus from the surface 
 with the distance f of the source from the surface, with the 
 
 radius r, and with the index of refraction n, is ~ = n l- 
 
 In this case all the lines are measured from the surface toward 
 the source of light. 
 
 If the surface is convex toward the source, a construction 
 shows that the focus may sometimes he real. The formula 
 for this case is obtained from the one just given by changing 
 the signs of those lines in it which are drawn from the surface 
 away from the source of light. 
 
 131. Lenses. A transparent body which is bounded by two 
 spherical surfaces is called a lens. A line drawn through the 
 lens, perpendicular to both its bounding surfaces, may be called 
 its axis. If the source of light is placed on the axis, refraction 
 will occur at the first surface and the refracted rays will pro-
 
 202 LIGHT 
 
 ceed. as if they were coming from or going toward the focus 
 formed by this surface. When they meet the second surface, 
 they will again be refracted, and their new directions will de- 
 termine a second focus. This focus is the focus of the source 
 formed by the lens. Its position may be found, as our descrip- 
 tion lias indicated, by using twice the formula for the focus 
 due to a single spherical surface. 
 
 There are two general types of lenses, called respectively 
 convex and concave lenses. In convex lenses the surfaces 
 which bound them are so shaped that the lens is thickest in 
 the middle; in concave lenses, the lens is thinnest in the 
 middle. It may be shown by construction and proved by 
 analysis, that the effect of the convex lens is to cause the rays 
 which fall upon it to converge, or at least to diverge less 
 widely than before they met the lens. On the other hand, the 
 effect of a concave lens is to diverge the rays more widely. 
 These classes of lenses are, therefore, called, respectively, con- 
 verging and diverging lenses. 
 
 The typical lens is the convex meniscus. The radii of the 
 two surfaces which bound it are both drawn toward the source 
 of light, and the radius of the second surface is less than that 
 of the first. If we suppose the lens to be so thin that we can 
 neglect its thickness in comparison with the other distances 
 involved, we may obtain a simple formula for the distance f 
 of the focus from the lens in terms of the distance f of the 
 source from the lens, of the radii r and s of the first and sec- 
 ond surfaces respectively, and of the index of refraction n of 
 
 the substance of the lens. This formula is 1 J 1 e =^(nl)( 1 ~\. 
 
 \r s I 
 
 In this formula all the lines are measured from the lens 
 toward the source. To adapt it to any other form of lens we 
 change the signs of those lines which are drawn from the lens 
 away from the source of light. The sign of f is left unchanged, 
 this quantity being the unknown term in the formula. If for 
 given conditions its sign is positive, the focus is on the same 
 side of the lens as the source of light, and is therefore a vir- 
 tual focus. If it is negative, the focus lies on the other side 
 of the lens and is a real focus.
 
 I.KIIIT. 203 
 
 If the source of light is at an infinite distance, the focus 
 obtained is called the principal focus, and its distance from 
 the lens, the focal length of the lens. It is plain that the 
 source and focus may be interchanged. The points thus related 
 are therefore called conjugate foci. 
 
 We may illustrate the use of the formula by applying it to 
 the double convex lens, the ordinary burning glass. In this 
 lens we must set the radius of the first surface negative, since 
 it is drawn away from the source of light. When the source 
 is sit infinity the focal length of the lens is the reciprocal of 
 
 (n 1) I - + - I and is negative, so that light falling from 
 
 an infinite distance on such a lens converges to a real focus. 
 As the source moves up toward the lens, the focus moves away 
 from it, until, when the source is at a distance from the lens 
 equal to its focal length, the focus is at an infinite distance 
 from it. If the source moves still nearer to the lens, the sign 
 of f becomes positive, showing that the focus is on the same 
 side of the lens as the source, or is now a virtual focus. 
 
 132. Images. If a small extended object stands transverse 
 to the axis of the lens, the light coming from any point of it 
 will come to a focus on the line drawn from it through the 
 centre of the lens. The assemblage of foci found in this way 
 for the different points of the object is its image. By con- 
 struction we may show that real images are inverted, while 
 virtual images are erect. The ratio between the height of the 
 object and the height of the image is equal to the ratio be- 
 tween their respective distances from the lens. 
 
 133. The Camera. The Eye. If a double convex lens is 
 placed in an aperture made in the wall of a chamber or 
 camera, images of external objects will be brought to a focus, 
 with more or less distinctness, on the opposite wall. By 
 using a movable screen, the image of any particular object can 
 be brought to exact focus on it. Such a screen is used in the 
 ordinary photographic camera. If the screen cannot be 
 moved, as in some forms of camera, the lens is so selected as 
 to give sharp images of objects at a certain distance. The
 
 204 LIGHT. 
 
 images of objects at a less distance are not sharp, and clear 
 pictures of them cannot be obtained. The images of more 
 distant objects are also not sharp, but the indistinctness in 
 them is much less marked. It is plain that, if we had at our 
 disposal a number of lenses of different curvatures, or if we 
 were able to modify the curvature of our lens, we might 
 obtain a distinct image of any object, even on a fixed screen. 
 .The eye is essentially a camera of this sort. Light which 
 enters through the pupil falls upon the crystalline lens, and 
 is brought to a focus upon the retina. The optic nerves are 
 distributed over the retina, and are affected by the image 
 which falls upon them. By a special muscle which controls 
 the crystalline lens, its shape is modified in order to produce 
 a sharp image, on the retina, of the particular object which is 
 under observation. 
 
 The angle subtended at the eye by the rays coming from 
 the extreme points of an object is called the visual angle sub- 
 tended by that object. It is evident that the perception of 
 small details depends upon the visual angles subtended by 
 them, and hence that they will be better perceived the nearer 
 they are to the eye. We cannot, however, push the examin- 
 ation of details to an extreme by bringing the object very 
 near the eye; for, as the object is brought nearer, its image 
 tends to recede behind the retina, and it can only be kept on 
 the retina by increasing the sphericity of the lens. This can- 
 not be done beyond certain limits, and after the object has 
 approached the eye within a certain distance, it can no longer 
 be seen distinctly. Even when it can be seen distinctly, the 
 effort involved soon tires the eye. The normal nearest dis- 
 tance at which objects can be seen distinctly without per- 
 ceptible effort is called the distance of distinct vision. 
 
 The nearsighted eye is one in which the crystalline lens 
 cannot be flattened sufficiently. Because of its too great 
 sphericity, it throws the images of distant objects in front of 
 the retina. This defect is corrected by placing before the eye 
 a concave lens, which slightly increases the divergence of the 
 light coming from distant objects, and so throws back their
 
 LIGHT. 205 
 
 images on the retina. The far-sighted eye is one in which the 
 crystalline lens cannot be made sufficiently curved. Because 
 of its flatness, it throws the images of near objects behind the 
 retina. This defect is corrected by placing before the eye a 
 convex lens, which lessen* the divergence of the liijht coming from 
 near objects, so that distinct images of them are cast on the 
 retina. 
 
 134. Optical Instruments. By viewing an object through 
 a lens or a combination of lenses, its visual angle may be in- 
 creased, while at the same time the light coming from it can 
 be properly brought to a focus on the retina of the eye. Such 
 combinations are called optical instruments. In general they 
 may be said to be used for magnifying objects. The ratio of 
 the visual angle of the object when seen through the instru- 
 ment to its visual angle when seen with the unaided eye 
 may be taken as a measure of the magnifying power of the 
 instrument. 
 
 The convex lens is used as a magnifying glass. If an object 
 which we wish to examine is placed in front of the lens, at a 
 distance from it somewhat less than its focal length, the rays 
 from the object will be rendered only slightly divergent by a 
 passage through the lens, so that an eye which receives them 
 after they have passed the lens can bring them to a focus on 
 the retina. These rays appear to the eye to come from the 
 virtual image of the object. By properly adjusting the dis- 
 tance between the lens and the object, this virtual image may 
 be brought to the position of distinct vision. When examined 
 by the eye it appears erect and larger than the object. The 
 amount of magnification depends on the shape of the lens. It 
 increases as the focal length of the lens decreases. 
 
 The telescope, in its first form, was constructed by a Dutch 
 optician, Jansen, who was led to it by an accidental observa- 
 tion made by his children. In the same form it was invented 
 by Galileo, and is generally known by his name. The opera 
 glass is an example of this form of telescope. It consists 
 essentially of a convex lens, or object glass, and a concave 
 lens, or eye piece, carried in a tube whose length 'may be ad-
 
 20(5 LIGHT. 
 
 justed. The distance between the lenses is less than the focal 
 length of the object glass. When the rays from a point of a 
 distant object pass through the object glass, they are made 
 to converge. Their convergence is so rapid that the unaided 
 eye cannot bring them to a focus on the retina. But, by 
 passage through the concave lens, they are made to diverge 
 slightly, so that the eye looking toward that lens is able to 
 bring them to a focus on the retina. The light therefore 
 enters the eye as if it came from a point situated at the apex 
 of the cone of rays which enters the pupil from the real point 
 of the object, after passing through the lenses. The eye 
 therefor* sees an image of the object. By a suitable adjust- 
 ment of the distance between the lenses, the image, which i? 
 virtual and erect, may be set at any apparent distance desired. 
 The construction will show that it subtends a larger visual 
 angle than the object itself. 
 
 The astronomical telescope, in its simplest form, contains a 
 convex lens, as object glass, and a convex lens, as eye piece. 
 The rays from a distant object form a real inverted image of 
 it by passage through the object glass. After passing through 
 the points of tht image, they proceed as diverging rays. If 
 they are then intercepted by the second convex lens or eye 
 piece, they will behave exactly as the rays do which come 
 from the points of an object that is examined by the magnify- 
 ing glass. On looking toward the eye piece, the eye will see a 
 virtual image of the real image formed by the object glass. 
 The visual angle subtended by this image is greater than that 
 subtended by the object. Since the real image examined by 
 the eye piece is inverted as respects the object, the virtual 
 image seen is also inverted. This circumstance is of no conse- 
 quence in astronomical observation. In instruments of this 
 sort, designed for terrestrial observation, a pair of lenses is inter- 
 posed between the other two, by which the real image is 
 inverted so as to be erect. The virtual image seen is then 
 also erect. 
 
 The microscope is also a combination of two convex lenses. 
 The object examined by it is placed just outside the principal
 
 LIGHT. 207 
 
 focus of the object glass, and the enlarged inverted image of 
 it, formed by the object glass, is examined by the convex lens 
 forming the eye piece. The magnified image in this case is 
 inverted. 
 
 135. Lenses of Large Aperture. The formulae which have 
 been given and the statements which have been made, with 
 respect to lenses, hold true when only a small portion of the 
 lens around its axis is used, and then as a first approximation. 
 When lenses of large aperture are used, the light from a 
 source on the axis does not come to an exact focus. The 
 intersection of the various rays determines an especially 
 illuminated surface, called the caustic surface. 
 
 Furthermore, in the case of such lenses, the dimensions of 
 the image are not in the same proportion to each other as 
 those of the object. These defects of the image are said to 
 be due to the spherical aberration of the lens. By combining 
 two lenses, so shaped that the spherical aberration of one is 
 in the opposite sense to that of the other, a single object glass 
 or eye piece can be constructed which is free from spherical 
 aberration. 
 
 136. Intensity of Light. By the intensity of light is meant 
 its illuminating power, as judged by the eye on observation 
 of the illumination of some standard surface. It is measured, 
 or rather two intensities are compared, by an instrument 
 called the photometer. One of the earliest forms of photo- 
 meter, invented by Count Rumford, is made by setting up a 
 vertical rod at a little distance from a white screen. The 
 two sources of light to be compared are set so as to cast 
 shadows of the rod near each other on the screen. The space 
 covered by the shadow from one source is illuminated by the 
 light from the other source. The sources are then moved 
 about until the two shadows appear equally illuminated, and 
 it is then said that the intensity of the light from the two 
 sources is the same. By comparing the effects of the sources 
 when they are set at different distances from the screen, it is 
 found that the intensity of the light from a source varies 
 inversely with the square of the distance from the source.
 
 208 LIGHT. 
 
 137. The Velocity of Light. In our work, up to this time, 
 we have treated the rays of light as if they marked paths 
 along which light travels from its source. No proof has been 
 given to justify this, nor is any needed, so long as we adhere 
 to our original conception of light, as being something which 
 originates at external bodies and reaches the eye from them. 
 Very different ideas prevailed among the older students of 
 optics about the velocity with which light travels. Descartes 
 thought that its velocity was infinitely great. Galileo, on 
 the other hand, conceived of it as being possibly so small that 
 it could be detected by a simple experiment. He proposed that 
 two observers, furnished with lanterns, should occupy two 
 stations at a considerable distance from each other. The first 
 observer was to expose his lantern. The second observer, see- 
 ing the light from it, was to expose his lantern in turn, and 
 the light from it was to be observed by the observer at the 
 first station. From the time which elapsed between the ex- 
 posure of the first lantern and the reception of light from the 
 second, Galileo hoped to determine the velocity of light. 
 When the experiment was tried, it was found that no per- 
 ceptible time elapsed between these two events, except that 
 which was unavoidably taken by the second observer in 
 receiving the sensation of light from the first lantern and in 
 exposing his own. So far as this experiment went, the velocity 
 of light was infinite. 
 
 It was proved not to be infinite, and its value was deter- 
 mined, by the Danish astronomer Roemer, in 1676. Roemer 
 had been engaged in studying the revolutions of Jupiter's 
 satellites. In the course of their revolutions, these satellites 
 are often eclipsed by passing into Jupiter's shadow, and the 
 times of these eclipses can be very exactly noted. From a 
 succession of such observations, Roemer found the period of 
 revolution of one of the satellites, and predicted the moments 
 of its eclipses. On continuing his observations, he found that 
 the actual times of the eclipses gradually departed from the 
 predicted times. When the observations were analyzed, it was 
 found that, when the earth was moving away from Jupiter,
 
 LIGHT. 209 
 
 the period of the satellite appeared longer, and when the earth 
 was moving toward Jupiter, shorter, than the mean or average 
 value of all the observed periods. Thus, if the times of eclipse 
 were predicted from observations made when the earth was 
 nearest Jupiter, the observed times when the earth was far- 
 thest from Jupiter would lag behind the predicted ones. Roe- 
 iner concluded that these observations could be best explained 
 by supposing that light travels with a finite, though very great, 
 velocity. On the basis of his observations, he stated that light 
 took 22 minutes to travel over the diameter of the earth's 
 orbit. 
 
 This conclusion of Roemer's was for many years uncon- 
 firmed. At length, in 1729, the English astronomer Bradley 
 made a discovery which seemed to confirm it. From long con- 
 tinued observations of the positions of the fixed stars, he found 
 that the stars apparently describe small paths or orbits, which 
 are completed in one year. The paths which those stars de- 
 scribe which lie near the plane of the earth's orbit are short 
 straight lines. Those of the stars which lie farther away from 
 this plane are ellipses, which approach circles more nearly as 
 the star lies farther from that plane. In Bradley's time it 
 was commonly thought that light consisted of streams of 
 minute particles, .shot out in straight lines from the luminous 
 body. On this theory of light, it was easy to explain Bradley's 
 observation. For, the velocity of the light particles entering 
 the eye would be the resultant of the velocity of light in space, 
 and of a velocity equal to that of the earth in its orbit and in 
 the opposite direction: and the direction in which the light 
 particle would reach the eye would be the direction of this 
 resultant. When the earth was moving across the path of the 
 light, the direction of this resultant would be such that the 
 star would appear displaced from its true position in the direc- 
 tion of the earth's motion. From the amount of the deviation 
 and the known velocity of the earth in its orbit, the velocity of 
 light could be calculated. The result obtained was in good 
 agreement with that previously obtained by Roemer's method 
 as improved by later observers. The phenomenon discovered by 
 Bradley is called the aberration of light.
 
 210 LIGHT. 
 
 In subsequent years the velocity of light has frequently 
 been measured by methods which do not involve astronomical 
 observations, and the values obtained by Roemer's and Brad- 
 ley's methods are confirmed. The methods employed for this 
 purpose will subsequently be discussed. While Bradley's con- 
 clusion was thus confirmed, the particular argument by which 
 he reached it has long ago been discarded, because of the aban- 
 donment of the theory upon which that argument was based. 
 The explanation of aberration by means of the theory of light 
 now universally accepted is still under discussion. There can 
 be no doubt that the aberration depends on the velocity of 
 light in the way Bradley assumed it did, but it is extremely 
 difficult to reconcile aberration with other results of experi- 
 ment. 
 
 138. Composition of White Light. The rainbow naturally 
 attracted the attention of the early students of optics. It was 
 soon perceived that the light which reaches the eye from the 
 bow is light from the sun, which has been reflected by the drops 
 of rain. Descartes was able to explain the size of the bow by 
 showing that, though light will reach the eye from drops in 
 any position after one internal reflection, yet that the light 
 that comes from those drops situated in the direction from 
 the eye in which the bow appears to be, will be much less 
 widely divergent, or, as he expressed it, will be denser, than 
 the light from drops in other directions. The colors of the 
 bow were thought t<> h<> swndnry ph^noim-mi. ISiniiliir colors 
 were observed in light reflected from drops of water, and in 
 the light which had passed through a prism or a lens. White 
 light was supposed to be the simplest sort of light, and the 
 colors produced by passage through bodies were supposed to be 
 caused by modifications impressed by those bodies upon white 
 light. 
 
 It was shown by Newton, in 1672, that this conception of 
 white light was entirely erroneous. Newton received a beam 
 of sunlight, which had passed through a small hole in a win- 
 dow shutter, upon a prism, and cast the emergent light on a 
 screen. He found that the light on the screen, instead of being
 
 LIGHT. 211 
 
 a circle, was a long colored strip. Some of the colors were 
 more deviated than others. The colored strip he called the 
 spectrum. The colors noted by him, beginning with the one 
 which is least deviated, were red, orange, yellow, green, blue, 
 indigo, and violet. We may interpret this experiment by say- 
 ing that the index of refraction of glass for these different 
 colors is different, and that white light, instead of being simple, 
 contains all these colors, and is analyzed into them by the 
 prism because of their different refrangibilities. Newton 
 proved that the refractive index was different for the different 
 colors, by placing another prism behind a small hole cut in the 
 screen upon which the spectrum was cast, and by turning the 
 first prism so as to throw the different colors successively 
 through the hole. When this was done, each color in turn was 
 refracted, without any modification of color, and the deviation 
 was different for each. To confirm this conclusion Newton 
 superposed the colors of the spectrum by the aid of other 
 prisms or of mirrors, and in other ways, and found that in 
 each case white light was again obtained. 
 
 As we now use the prism, white light is passed through a 
 narrow slit, which stands parallel to the edge of the prism, and 
 a lens is placed in the path of the light, so that after it has 
 passed through the prism, the slit is sharply focussed on the 
 screen. Each different color which exists in white light under- 
 goes its peculiar refraction in the prism, and the spectrum 
 obtained is really a row of differently colored images of the 
 slit. When the slit is very narrow, these images overlap very 
 little, and the different colors are obtained with as little ad- 
 mixture of others as possible. Such a spectrum is called a 
 pure spectrum. 
 
 The separation of white light into its constituent colors by 
 refraction is called dispersion. The length of the spectrum 
 obtained, compared with the deviation of the whole spectrum, 
 is called the dispersive power of the prism. 
 
 139. Chromatic Aberration. In view of the composite 
 nature of white light, and the different retfrangibilitiee of its 
 constituent colors, it is evident that the focus of a point
 
 212 LIGHT 
 
 source, from which white light comes, will be different for each 
 of the constituent colors, and hence that the image of an object 
 viewed through a lens will be colored around the edges. Thia 
 defect of the image is said to be due to chromatic aberration. 
 From a few experiments which he made, Newton concluded 
 that the dispersive power of a prism was proportional to its 
 refractive power, so that if two prisms, or two lenses, were 
 superposed in such a way as to annul the dispersion, they 
 would at the same time annul the refraction. Newton there- 
 fore decided that no combination of lenses could be made which 
 would give an image free from color. He accordingly con- 
 structed a concave spherical mirror to be used instead of the 
 object glass in an astronomical telescope. The law of reflec- 
 tion being the same for all colors, the image formed by this 
 mirror was free from color. Telescopes of this sort were much 
 superior to the refracting telescopes then in use, and for many 
 years quite superseded them. 
 
 In 1757, Dollond, an English optician, investigated again 
 the relation between the dispersive and refractive powers of 
 different kinds of glass, and found that no such proportion be- 
 tween them prevailed as Newton had supposed. This being so, 
 it became possible to construct a combination of two lenses, 
 one of flint glass and the other of crown glass, so shaped as to 
 correct the dispersion and yet capable of refracting light and 
 forming an image. Such a combination is called an achro- 
 matic combination. When an achromatic combination is used 
 as the object glass of a telescope or microscope, the image 
 formed by it is almost entirely free from color at the edges, 
 and by the use of another achromatic combination as eye 
 piece, the disturbing effects of dispersion are almost entirely 
 avoided. 
 
 The dispersive power of different materials for different 
 colors does not follow any general law, though the relative 
 dispersive powers for the different colors are nearly the same. 
 The achromatic combination of two lenses can therefore only 
 completely correct for two colors. The image formed by it is 
 still slightly colored, owing to this so-called irrationality of
 
 LIGHT. 213 
 
 dispersion. By adding a third lens, a further correction can 
 be made. In the finest microscope lenses, made after the de- 
 signs of Abbe, the corrections for both spherical and chromatic 
 aberration have been pushed so far that the lenses are practi- 
 cally perfect. 
 
 140. Colors of Thin Films. The attention of Newton was 
 directed to the colors seen in thin films, such as sheets of mica, 
 or the walls of a soap bubble. These colors had been already 
 investigated by Hooke. Newton's most important observations 
 were made by the help of the film of air between a plane sheet 
 of glass, and a slightly convex lens laid upon it. When light 
 was allowed to fall nearly perpendicularly upon this film, and 
 the eye was placed above it, a succession of colored rings was 
 seen, having as a common centre the point of contact between 
 the two pieces of glass. At this point of contact there was a 
 black spot. Newton showed that when light of one color was 
 used, the rings were of the same color, and that the diameter 
 of the successive rings was different for the different colors. 
 The diameter of the ring of any order was greatest when red 
 light was used, and least when violet light was used. When 
 light of one color was used, the intervals between the rings 
 were black. From this observation it was easy to explain the 
 colors seen with white light. At any one place in the film some 
 of the constituents of the white light will not appear, or will 
 be extinguished, and the color actually seen is due to the mix- 
 ture of the remaining constituents, which are not extinguished. 
 
 From the known curvature of the lens under which the 
 film was formed, Newton calculated the thickness of the film 
 at different distances from the black spot at the centre of the 
 system of rings, and found that the rings* of any one color 
 appeared at parts of the film, whose thicknesses were in the 
 proportion of the numbers 1, 3, 5, 7, etc. This result indi- 
 cated some sort of periodicity, or regular alternation in the 
 properties of the light. 
 
 When the film is all one thickness, the color seen in it is 
 the same in all parts, except for slight modifications due to 
 the different angles of incidence upon the film of the light 
 from the source.
 
 214 LIGHT. 
 
 141. Diffraction. Other systems of colored bands of light 
 were first observed by Grimaldi, and were also observed by 
 Newton. They occur when light, which has passed through 
 a very small opening, passes the edge of an obstacle, or 
 through another small opening. They are best observed if the 
 first opening is a narrow slit. In this case, if the straight 
 edge of a screen is placed in the light coming through the slit 
 and parallel with it, the shadow of this edge cast on a re- 
 ceiving screen is bordered with three or four narrow colored 
 bands, standing outside the shadow. The edge of the shadow 
 itself is not sharp, as it should be if light traveled in geo- 
 metrically straight lines. The line on the screen in which it 
 would be met by a plane passing through the slit and the edge 
 of the obstacle is called the edge of the geometrical shadow. 
 In the conditions described, the colored bands stand outside 
 the edge of the geometrical shadow, and the illumination ex- 
 tends within it, gradually fading out until it becomes imper- 
 ceptible. 
 
 If a narrow slit is placed in the path of the light and 
 parallel with the first slit, the light which comes through it is 
 a narrow band of white light, bordered on both sides by a 
 succession of bands of variously colored light. 
 
 The phenomena here described and others like them, are 
 said to be due to the diffraction of light. 
 
 142. Double Refraction. The peculiar phenomena which 
 appear when a ray of light falls upon a crystal of calcite or 
 Iceland spar were described in 1669, by Bartholinus. The light 
 undergoes a double refraction. That is, it is broken by re- 
 fraction into two rays, which have different directions in the 
 crystal. If the incident ray is perpendicular to the surface, 
 one of the refracted rays is also perpendicular to the surface. 
 As the incident ray is differently inclined to the normal, this 
 refracted ray follows the ordinary law of refraction. It is 
 therefore called the ordinary ray. The other ray, whose law 
 of refraction .Bartholinus could not discover, is called the 
 extraordinary ray. If a surface is cut on the crystal per- 
 pendicular to the axis of the crystal, the incident ray per-
 
 LIGHT. 215 
 
 pendicular to that surface is not doubly refracted. The axis 
 thus determined is called the optic axis. 
 
 Huygens investigated the double refraction of Iceland 
 spar, and discovered the law by which the refraction of the 
 extraordinary ray may be described. As the description of 
 this law depends upon the theory of light which Huygens 
 used in obtaining it, we shall postpone the consideration of it 
 until we discuss that theory. 
 
 Huygens discovered that other crystals possess the prop- 
 erty of doubly refracting light, and subsequent investigations 
 have shown that all crystals, except those of the isometric 
 system, are doubly refracting. 
 
 143. Polarization of Light. In his investigation of Iceland 
 spar, Huygens discovered another remarkable property of 
 light. To show it, we first allow a narrovr beam of light to 
 fall perpendicularly upon one face of a crystal of Iceland spar. 
 The light is then doubly refracted, and emerges from the 
 opposite face in two nearly parallel beams. If a second crystal 
 of spar is placed in the path of these beams, so that they fall 
 perpendicularly, or nearly so, upon one of its faces, each of 
 them will ordinarily be doubly refracted, and four beams of 
 light will emerge from the second crystal. A difference is 
 observable between the double refraction in this case, and 
 that observed in the first crystal, for whereas the two 
 emergent beams from the first crystal were of equal intensity, 
 the two beams into which each of them is divided by the 
 second crystal are, in general, not of equal intensity. Indeed, 
 by turning the second crystal around an axis parallel with 
 the beam of light, two positions of it may be found in which 
 one beam of each pair disappears altogether, while the other 
 beam of each pair bus the intensity of the beam before it en- 
 tered the second crystal. We may describe this phenomenon 
 otherwise by saying that the first crystal divides the incident 
 beam into an ordinary and an extraordinary beam of equal 
 intensities. The second crystal divides each of these into an 
 ordinary and an extraordinary beam, whose intensities are 
 generally unequal. One position of the second crystal can
 
 216 LIGHT. 
 
 be found for which the extraordinary beam of the first 
 ordinary beam, and the ordinary beam of the first extra- 
 ordinary beam, are suppressed or disappear. When the 
 crystal is turned another position can be found, in which 
 the ordinary beam of the first ordinary beam, and the 
 extraordinary beam of the first extraordinary beam, are 
 suppressed. The light which has been given these prop- 
 erties, by which it seems to differ in different directions in 
 space transverse to the beam, is said to be polarized. 
 
 Huygens could give no explanation of polarization, and its 
 exhibition by light which had passed through Iceland spar was 
 for a long time unique. It was shown much later, as we shall 
 subsequently consider more at length, that whenever double 
 refraction occurs, the two rays produced are always polarized, 
 and it was discovered also that light may be polarized by ordi- 
 nary reflection and refraction, or in other ways. 
 
 144. The Emission Theory of Light. We have now passed 
 in review the most important phenomena exhibited by light 
 which were known to the earlier workers in that subject, and 
 which had to be considered in any attempt to construct a theory 
 of light. From very ancient times there had been vaguely 
 enunciated two rival theories of light. In one of these, light 
 was supposed to be an actual emanation of some substance 
 from the luminous body. In the other, it was assumed that 
 the universe is pervaded by an intangible medium, and that 
 light consists in disturbances transmitted through that 
 medium. In the form which this latter theory took at the 
 hands of Huygens, these disturbances were supposed to be 
 waves, similar in their general features to the waves of sound 
 in air. Huygens, however, could not adapt this theory to the 
 explanation of the transmission of light in straight lines. 
 When the waves of the ocean enter the narrow opening of a 
 harbor, they do not proceed in a narrow band across the waters 
 of the harbor, leaving the water on either side of the band 
 undisturbed, but they spread out in all directions from the 
 entrance. In the same way, a sound which is made inside a 
 building will spread out in all directions when it passes
 
 LIGHT. 217 
 
 through an open window. This behavior of such waves as we 
 can actually experiment with seemed to Huygens' critics to 
 make the wave theory of light untenable, for there is nothing 
 more conspicuous about the behavior of light than its transmis- 
 sion in straight lines and the fact that it does not spread out 
 in all directions after it has passed through an opening. 
 Largely on this account Newton refused to consider the wave 
 theory, and set himself to develop its rival, the emission theory 
 of light. A brief description of this theory, and a comparison 
 of it with the wave theory, by which it was finally displaced, 
 will furnish an excellent illustration of the methods of reason- 
 ing used in constructing a physical theory. Newton assumes 
 that a luminous body shoots out from itself small bodies, which 
 he calls corpuscles. The velocity of these corpu-rlcs is that of 
 light. Since the corpuscles are masses, they proceed in 
 straight lines, by the fundamental law of inertia, unless they 
 encounter some body by which their motion is changed. By 
 supposing them perfectly elastic, it is easy to show that when 
 the corpuscles fall obliquely on a smooth surface, they will 
 rebound from it in a direction which obeys the law of reflec- 
 tion. By supposing further that the corpuscles are sometimes 
 strongly attracted by ordinary matter, when the distance be- 
 tween the corpuscle and the matter is exceedingly small, and 
 that the corpuscles, having entered the matter, can proceed 
 through it without interruption, Newton shows that the path 
 of the corpuscle will be changed in direction on its entering 
 the mass of matter in accordance with the law of refraction. 
 As a necessary consequence of this assumption, the velocity 
 of a corpuscle is greater in the more highly refracting medium. 
 To account for the different colors in white light, Newton 
 assumes that corpuscles of different mass and perhaps of other 
 different properties correspond to them. The colors of thin 
 films render it necessary to make some additional hypothesis 
 which will impart to the corpuscles some sort of periodicity. 
 The particular form of this hypothesis which Newton makes 
 assumes the existence of an elastic medium, called by Newton 
 the ether, in which waves are set up whenever a corpuscle
 
 218 LIGHT. 
 
 strikes the surface of a body. These waves travel along with 
 the corpuscle, if it is reflected, and with nearly the same 
 velocity, so that when it meets another surface the wave can 
 be thought of as either pushing it forward to help it pass 
 through the surface, or as pulling it backward to prevent its 
 passing through. Newton names these two conditions the 
 fits of easy transmission and of easy reflection. By the help 
 of these fits the light in the thin film acquires the necessary 
 periodicity. Another and an easier form of the same hypothe- 
 sis is that given by Boscovich, who assumes that the corpus- 
 cles are in continual rotation around axes perpendicular to 
 their paths, and that their ends are different, so that as the 
 corpuscle reaches a surface it will be reflected if one of its 
 ends is presented to the surface, and will be transmitted if 
 the other end is presented to the surface. In order to explain 
 the bands produced by diffraction, Newton makes the further 
 supposition that the corpuscles which pass near a solid edge 
 move back and forward, as he describes it, with a motion like 
 that of an eel, and so proceed in one direction or another 
 according to the direction in which they are moving when they 
 escape the influence of the edge. Newton's ingenuity fail* 
 him when he attempts to apply this theory to explain double 
 refraction, and he cannot explain polarization further than 
 by saying that it shows that light has sides. In the subse- 
 quent development of the emission theory, which held its 
 ground for a century and a half, additional hypotheses were 
 made to account for the other phenomena which were dis- 
 covered. It cannot escape notice that this theory, which at 
 the outset is so simple, and which accounts so well for recti- 
 linear transmission, and for reflection and refraction, requires 
 a new hypothesis for each additional phenomenon which is to 
 be explained. Each hypothesis is of service only in explaining 
 the phenomenon which leads to its adoption, and when a new 
 phenomenon is considered, a new hypothesis has to be made to 
 explain it. When the consequences of any of these hypotheses 
 are followed out, no new and previously unknown truths 
 about light are discovered. The only necessary conclusion
 
 LIGHT. 219 
 
 from the fundamental hypotheses of the theory which was 
 an addition to the knowledge of light, turned out to be false 
 when it was examined by experiment. As has been stated 
 already, Newton's principles led to the conclusion that light 
 travels faster in optically denser bodies. This conclusion was 
 tested experimentally by Foucault, in 1850, and found to be 
 erroneous. Light really travels slower in optically denser 
 bodies. 
 
 145. The Wave Theory of Light. The wave theory of light 
 received its first impulse from Huygens, who laid down the 
 fundamental principle upon which its development depends. 
 Huygens perceived that if a train or succession of waves is 
 passing through a medium, the disturbance at any point in 
 the medium does not depend solely on a disturbance coining 
 directly to it from the source, but is the resultant of all the 
 disturbances which reach it at the same time from all the 
 disturbed parts of the medium. To put it otherwise, if a point 
 in the medium is disturbed by a wave, it thereby becomes the 
 centre of a wave disturbance, which goes out from it nearly 
 as if it were a centre of an original disturbance. If we then 
 consider a point in the medium, and draw a spherical surface 
 around it as centre, and suppose each of the points on that 
 surface to be in some way disturbed by a train of waves, each 
 of those points will send a wave toward the centre of the 
 sphere, and the disturbance at the centre will be the resultant 
 effect produced by the waves which reach it, at the same 
 instant, from the different points of the surface. In order to 
 meet the facts of the case, especially in order to explain that 
 a wave disturbance is not transmitted backward as well as 
 forward, Huygens had to assume that the elementary wave 
 sent out from a disturbed point does not transmit equal dis- 
 turbances in all directions, but that the greatest disturbance 
 is transmitted along the ray, in the direction in which the 
 wave is advancing which disturbs the point, and that the dis- 
 turbances transmitted in oblique directions diminish in in- 
 tensity until, in the opposite direction to that of the ray, they 
 fall off to nothing. The analysis by Kirchhoff and others of
 
 220 
 
 the mode of transmission of waves in a medium shows that 
 this hypothesis of Huygens is a sound one. The principle 
 which has been stated is known as Huygens' principle. 
 
 As a fundamental hypothesis of the wave theory, Huygens 
 assumed the existence of an ether, or universal medium, in 
 which the waves which constitute light are set up and trans- 
 mitted. This ether is not ordinary matter. It is not cogniz- 
 able by the senses. It occupies space that is otherwise void, 
 as well as space filled with ordinary matter. It seems to be 
 frictionless, at least to such a degree that slowly moving 
 bodies, like the planets, pass through it without loss of energy. 
 It possesses certain properties, analogous to the elasticity 
 and the density of matter, which are modified, but not de- 
 stroyed, by the presence of matter. As we shall see later, 
 these properties are best described as electric properties, but 
 we may, for the present, use the material analogy. Huygens 
 supposed that waves can be set up and transmitted in this 
 ether according to the principle already described, and then 
 proceeded to give explanations of certain well-known facts. 
 These explanations we shall now consider. 
 
 Huygens explained the rectilinear transmission of light 
 through an opening, by drawing the elementary waves around 
 the points which stand in the line of the opening, and showing 
 that directly beyond the opening these waves have a common 
 envelope, while on either side of the opening they have not. 
 He supposed the only effective part of the resulting dis- 
 turbance to be that caused by this common envelope. This 
 explanation, though ingenious, is not sufficient, and rectilinear 
 transmission was not really explained by Huygens' form of 
 the wave theory. 
 
 To give Huygens' explanation of reflection and refraction, 
 we make the following construction. We draw the straight 
 line GH to represent the intersection of the plane of the paper 
 with a plane reflecting surface at right angles to it. We con- 
 sider a series of waves advancing obliquely toward this sur- 
 face in air, forming a beam, which is bounded by the extreme 
 rays EA and FC. At a certain instant, one of the wave fronts
 
 LIGHT. 21 
 
 represented by the line AB will meet the surface at the point 
 A. By Huygens' principle that point will become the centre 
 of a wave disturbance, which proceeds from it in both media. 
 We shall consider it first in the second medium. At the end 
 of a time in which the disturbance which was at B has reached 
 the point C, the wave from A has become a sphere with A as 
 its centre. We shall suppose that the velocity of light in the 
 second medium is less than that in the first, so that the radius 
 of this sphere is less than the line BC. If we consider the light 
 coming from any point between the points A and B, we see 
 that it will reach the surface GH, at some point between A 
 and C, after part of the above mentioned time has elapsed, 
 and that there will proceed out from that point a spherical 
 wave. W T hen all such spherical waves are constructed, it will 
 be found that they have a common tangent, which is a line 
 drawn from the point C. This line is therefore the envelope 
 of all the elementary waves which correspond to the original 
 disturbances in the wave front AB, and therefore represents 
 the refracted wave. If we draw from the point A the radius 
 AD to the point of tangency, the direction thus determined is 
 the direction of the refracted beam. The lines BC and AD are 
 distances passed over by the reflected and refracted waves 
 respectively in the same time, and they are therefore propor- 
 tional to the velocities of light in the first and second media. 
 From the construction we see that these same lines are pro- 
 portional to the sines of the angles of incidence and refraction. 
 The ratio of these sines is therefore equal to the ratio of the 
 velocities of light in the two media, and this ratio is pre- 
 sumably constant for the two media. We therefore deduce in 
 this way the law of refraction. Huygens' hypothesis that the 
 velocity of light is less in the denser medium, which is a 
 necessary hypothesis of the wave theory, was confirmed by 
 the experiments of Foucault already referred to. 
 
 A construction that is essentially similar, in which the 
 elementary waves from the points on the lines A C are drawn 
 in the upper medium, will lead to the law of reflection. 
 
 By the use of the wave theory, Huygens was able to give
 
 222 LIGHT. 
 
 a description of the double refraction in Iceland spar. To do 
 so he made the supposition that the disturbance which reaches 
 a point on the surface of the spar sets up two waves, one of 
 which is spherical, and corresponds to the ordinary ray. The 
 other is an ellipsoid of revolution, having its axis of revolu- 
 tion parallel to the axis of the crystal. By a construction 
 which is essentially similar to the one we have used in ex- 
 plaining ordinary refraction, Huygens was able to determine 
 the direction of the extraordinary ray for various incidences, 
 and to show that, in each case, it was in complete accord with 
 his observations. 
 
 Huygens did not develop the wave theory beyond this 
 point. In the form in which he gave it, it was really defective 
 in that it did not explain satisfactorily rectilinear trans- 
 mission. The emission theory, therefore, svipported by the 
 great authority of Newton, remained prevalent until another 
 set of phenomena was discovered, with which the development 
 of the wave theory really begins. 
 
 146. Interference of Light. In 1803 Young discovered an 
 effect produced by light, which of itself, and because of the 
 consequences deduced from it, was almost conclusive in favor 
 of the wave theory. 
 
 Young allowed a beam of light to pass through a very 
 small opening, and received it on a distant screen. On inter- 
 posing a narrow strip of card in the path of the beam, and 
 observing its shadow on the screen, he perceived that it was 
 not uniform, but contained a number of alternately light and 
 dark bands, parallel with its edges. He showed that these 
 bands were due to the combination of the two portions of the 
 beam which had passed the two edges of the card; for, on 
 cutting off the light on one side of the card with another 
 screen, the bands disappeared. 
 
 Young subsequently described another form of this experi- 
 ment, which presents its essential features very clearly. In 
 it, light which has passed through a very small opening is 
 allowed to fall on two small holes 'or parallel slits set close 
 together. On observing the light cast through these holes on
 
 LIGHT. 223 
 
 a screen behind them, the region in which the two beams of 
 light overlap is seen to be crossed by a set of parallel light 
 and dark bands. When either of the holes through which the 
 light comes is covered, these bands disappear. They cannot 
 be obtained when the light which falls on the openings comes 
 directly from an ordinary source. This experiment is known 
 as Young's experiment. 
 
 Young explained these phenomena by the help of the wave 
 theory. To explain the second experiment, we suppose that 
 the waves, starting from the first opening, which we shall 
 hereafter call the source, set up waves at each of the two 
 openings in the screen, these waves proceeding from the open- 
 ings almost as if they were independent sources. If we con- 
 sider the effect at any one point on the receiving screen pro- 
 duced by both these waves, we see by construction that it 
 will depend on the relative distances of the point from the 
 two openings. When the difference of these distances is equal 
 to half a wave length, or any odd multiple of half a wave 
 length, the two waves will reach the point in opposite phases, 
 and by the principle of the superposition of small motions, 
 the resultant effect will be zero. The black bands that are 
 seen correspond to this condition. At intermediate points 
 between the black bands, for which the difference of the dis- 
 tances to the openings is an even number of half wave lengths, 
 the waves from the openings will meet in similar phases. They 
 will therefore increase the light received at those points. 
 
 We explain the first experiment in an essentially similar 
 way, considering that the two beams of light which combine 
 to produce the light and dark bands are those portions of the 
 original beam which pass close to the edges of the card and 
 are turned into its shadow by diffraction. 
 
 The most striking single argument that was ever adduced 
 in favor of the wave theory was that presented by the disap- 
 pearance of the system of bands when the light from one of 
 the openings was intercepted, and their reappearance when it 
 was again allowed to pass. This effect follows as a natural 
 consequence of the wave theory and is inconsistent with any
 
 224 LIGHT. 
 
 reasonable form of the emission theory, for it seems impossible 
 to explain how the meeting of two streams of particles will 
 produce an effect different from that which would be produced 
 by either stream of particles acting alone. 
 
 Waves which are superposed on each other, in the way 
 which has been "described, so as to produce variations of 
 intensity in different parts of the field through which they 
 are passing, are said to interfere, and the general phenomenon 
 is called the interference of light. The waves do not interfere 
 with each other in such a way as to destroy each other's in- 
 dividuality. Each wave passes through any point, and pro- 
 ceeds beyond it, as if no other wave had met it at that point. 
 This follows as a consequence of the principle of superposition. 
 All that is meant by interference is that the waves combine 
 or interfere with each other at any point as respects their 
 individual effects, so that the effect perceived is that due to 
 their combined action. 
 
 As soon as Young had discovered interference, he found that 
 with its help he could apply the wave theory to explain the 
 colors of thin films. Consider a train of waves incident on the 
 upper surface of a film. A normal to these waves will meet 
 that surface at a point A, and another normal, at the neighbor- 
 ing point B. The waves which reach A are partly reflected and 
 partly refracted into the film. The wave in the film proceeds 
 in the direction of its normal to the point C on the second face 
 of the film, where it will again undergo reflection within the 
 film, and refraction into the outside medium. The reflected 
 part will proceed again through the film, and part of it will 
 emerge at the point B. At that point it will be superposed on 
 a wave directly reflected there, and the effect produced in the 
 direction of the reflected wave normal will be due to the com- 
 bination of these waves. If they are in the same phase they 
 will produce an enhanced effect, so that light of the peculiar 
 color which corresponds to the wave length will be seen. If 
 they are in opposite phases, they will annul each other and no 
 light will be seen. When white light is used, which we may 
 suppose to contain waves of all periods lying between certain
 
 LIGHT. 225 
 
 limits, this description shows that some of its constituents will 
 be suppressed, while others will not be, so that the reflected 
 light will appear colored, and the color will differ for different 
 thicknesses of the film. 
 
 When we consider the relation between the thickness of the 
 film and the wave length of the light which is enhanced or 
 suppressed, it seems at first sight that the two waves from the 
 point B would be in similar phases when the distance AC+CB, 
 or twice the thickness of the film, is equal to the wave length, 
 or to any multiple of the wave length of light in the film. This 
 turns out to be not the case. In the conditions described no 
 light leaves the point B. To account for this Young assumed 
 that the reflection of light takes place, in different circum- 
 stances, with or without change of sign. At one of the sur- 
 faces, say at the upper one, the light is passing from a denser 
 to a rarer medium, and from the analogy of the reflection of 
 sound, we may suppose it to be reflected there without change 
 of sign. At the other surface it passes from a rarer to a 
 denser medium, and we accordingly suppose it to be reflected 
 there with change of sign. In this way the light which is re- 
 flected in the film has its phase reversed, and in considering its 
 combination with the light reflected on the upper surface, we 
 must consider its phase opposite to that which would be due 
 to the distance over which it travels in the film. With this 
 additional hypothesis, which subsequent analysis has shown to 
 be justified, Young was able to explain the colors of thin films, 
 and the size of the successive rings in Newton's experiment, 
 and to show that the wave lengths of the different colors, cal- 
 culated from the size of the rings, were in agreement with those 
 obtained by him from his original experiments on interference. 
 
 If we trace the course of the waves which emerge after 
 passing through the film, we find that, of the waves which are 
 superposed at the point D, one of them has passed through the 
 film once, while the other has passed through it three times, 
 and has besides undergone two reflections with change of sign. 
 The difference of phase of the waves superposed at D will there- 
 fore depend only on the difference of their paths in the film,
 
 226 LIGHT. 
 
 and light will emerge when the difference in path is equal to 
 an even number of half wave lengths in the film. Light is thus 
 transmitted through the film when its thickness is such that 
 no light is reflected by it. In Newton's experiment a system 
 of rings will be seen by transmitted light which is complemen- 
 tary to those seen by reflected light; that is, in which the 
 colors are so arranged that when they are superposed on the 
 reflected system, a uniform field of white light is the result. 
 
 A more extended study of the colors of thin films, in which 
 account is taken of the fact that the light which enters the 
 film does not all leave it after one or two internal reflections, 
 confirms the conclusions of the elementary theory in the case 
 of the reflected light, but shows that the transmitted light will 
 never be entirely extinguished for any thickness of the film. 
 As the thickness of the film changes, the transmitted light will 
 pass through a series of maximum and minimum values. These 
 conclusions are confirmed by observation. 
 
 Young also applied the wave theory to explain diffraction. 
 He supposed that the diffraction bands outside the shadow are 
 caused by the interference of the light which passes the obstacle 
 outside its geometrical shadow, with light reflected from the 
 edge of the obstacle. In this way he could explain in general 
 the production of the diffraction bands, but he could not ac- 
 count for their exact position, nor could he readily explain 
 some other of the diffraction phenomena. 
 
 147. Fresnel's Development of the Wave Theory. A few 
 years after Young's discovery of interference, Fresnel began 
 researches by which the wave theory of light was carried to a 
 very high development. His early experiments on interference 
 were similar to Young's, but he made an important step in 
 advance by demonstrating the interference of light in a way 
 that met an objection which had been raised to Young's experi- 
 ment. In regard to that experiment it was said that the black 
 bands observed by Young were not produced by simple inter- 
 ference, but were diffraction phenomena due to an action upon 
 the light of the edges of the opening through which it passed. 
 Fresnel avoided this objection by substituting for the two open-
 
 LIGHT. 227 
 
 ings two images of the source formed in two plane mirrors, 
 which met along one edge, and were very slightly inclined to 
 each other. By properly adjusting the inclination of these 
 mirrors, the two beams of light reflected from them were made 
 to overlap, and interference bands, similar to those observed by 
 Young, were produced. Fresnel found that by placing an eye 
 piece in the path of these beams, the interference bands could 
 be observed in it. He was thus able to observe them and to 
 measure their distances much more precisely than can be done 
 when they are allowed to fall on a screen. All the interfer- 
 ence and diffraction phenomena can be observed best in this 
 way. 
 
 It was evident that the wave theory, however successful it 
 might be in explaining some of the phenomena of light, could 
 not be accepted if it could not be made to account for the recti- 
 linear transmission of light. By a combination of Huygens' 
 principle with Young's principle of interference, Fresnel was 
 not only able to account for rectilinear transmission, but also 
 to explain by the same principles all the phenomena of diffrac- 
 tion. To describe Fresnel's explanation of rectilinear trans- 
 mission of a linear wave, we construct a diagram as follows: 
 Supposing the source of light to be a point at an infinite dis- 
 tance on the left, we draw a straight line to represent one of 
 the rays sent from it, and at right angles to this line we draw 
 a number of equidistant straight lines. The distance between 
 two of these lines is taken to be half a wave length. The suc- 
 cessive lines, therefore, represent parts of the successive waves 
 which are in opposite phases. Take a point on the ray and with 
 it as centre describe a circle. By the principle of Huygens 
 every point on this circle, being disturbed by the waves passing 
 over it, will act as a centre of disturbance and will send out a 
 wave. The waves from all the points on the circle will reach 
 the centre at the same instant, and their phases at that centre 
 will differ in the same way that they differ in the circle. The 
 effect at the centre will be that due to their superposition. 
 Now, if we examine those portions, or elements, of the circle 
 which are intercepted between the parallel lines of the figure,
 
 we see that the disturbances from two successive elements will 
 in general be in opposite phases, so that when they are super- 
 posed at the centre, they tend to destroy each other. If we at 
 first confine our attention to that portion of the circle which 
 lies nearer the source, we see at once that much the largest 
 element is that which lies nearest the line joining the source 
 with the centre of the circle, and that the successive elements, 
 as we go out around the circle, soon become appreciably equal 
 to each other. It is natural to suppose that the effect of any 
 one of these elements is proportional to its length. Those ele- 
 ments which are nearly equal to each other, therefore, annul 
 each other's effects in pairs, and the effect at the centre is due 
 to the very considerable preponderance of the effect of the first 
 element over that of those which lie outside of it. 
 
 If we consider that portion of the circle which lies farther 
 from the source than the centre, we shall find the same differ- 
 ence in the size of the successive elements as we found before. 
 We may, however, neglect the action of this part of the circle 
 if we remember that in the application of Huygens' principle 
 we have agreed to suppose that the effect produced by the ele- 
 mentary waves diminishes with the obliquity, and that in par- 
 ticular their effect vanishes in the direction opposite to that 
 in which the actual waves progress. 
 
 ' It thus appears thdt the effect which reaches the centre of 
 the circle is practically due entirely to that element of the 
 circle which lies nearest the straight line joining the source 
 with the centre. The light received at the centre, therefore, 
 appears to travel from the source in a straight line. The con- 
 clusion which we have drawn from this .description is fully 
 confirmed by analysis. 
 
 Fresnel applied similar principles to the explanation of the 
 diffraction produced by the edge of an obstacle. The linear 
 waves which cause the effect are perpendicular to the edge as 
 they pass the obstacle. To consider this case we construct the 
 diagram as before, and suppose, first, that the obstacle is so 
 interposed that the waves which would pass the circle are all 
 intercepted. As the obstacle is lowered, waves are set up at
 
 LIGHT. 229 
 
 the different portions of the circle in succession. Those from 
 the uppermost parts of the circle have no effect, on account of 
 the obliquity, but as the obstacle is still further lowered, waves 
 are at last set up in elements which send appreciable effects to 
 the centre of the circle. As we have already seen, the element 
 nearest to the obstacle will be the largest, and it will send an 
 effect to the centre that will not be entirely annulled by the 
 effects of the other elements. As the edge of the obstacle ap- 
 proaches the line from the source to the centre, the element 
 nearest to it becomes relatively larger and its effect relatively 
 greater. When the edge of the obstacle is on that line, the 
 effect at the centre is that produced by half the wave. The 
 wave theory, therefore, leads naturally to an explanation of the 
 illumination of the geometrical shadow. If the obstacle is still 
 further lowered, so as to expose the first element on the other 
 side of the line joining the source with the centre, the effect 
 of that element is added to that of the upper half of the circle 
 and the centre is more highly illuminated. If the obstacle 
 exposes two elements below the line, their effects at the centre, 
 being in opposite phases, partly destroy each other and the 
 centre is relatively darker. Similar alternations of relative 
 brightness and darkness occur at the centre as the successive 
 elements are exposed. The wave theory thus accounts for the 
 diffraction bands seen outside the geometrical shadow. 
 
 When Fresnel made a calculation of the positions of the 
 diffraction bands, using the wave length of light which he had 
 already obtained from interference experiments, he found that 
 the calculated positions agreed with the observed positions of 
 the bands. 
 
 We may consider also the diffraction produced by a narrow 
 slit. If we take a limited line whose length is the width of the 
 slit and move it across our diagram, the elements which it will 
 cover will be those exposed by the slit, and we may determine 
 the effect produced by the slit by considering the elements of 
 the circle which will be exposed. If, for example, the slit is so 
 placed as to expose the two central elements, the centre of the 
 circle will be brightly illuminated. If, on the other hand, it
 
 230 LIGHT. 
 
 exposes the first and the second element, they will partly coun- 
 teract each other and the centre will be relatively darker. If 
 it exposes three elements, the third will supplement the effect 
 of the first and the centre will be brighter again. In general 
 the centre will be illuminated if the slit exposes an odd number 
 of elements and will be dark if it exposes an even number. 
 
 On the natural assumption that the color of light depends 
 on its wave length, it is easy to explain the colors in the dif- 
 fraction bands. For, because of the difference of wave length, 
 the elements of our diagram will differ in length, for the differ- 
 ent colors, and a certain position of the slit which will produce 
 darkness at the centre for one wave length will not do so for 
 another. For different positions of the slit, therefore, if we 
 use white light, the centre will be illuminated with different 
 colors. Consideration of the diagram will make it plain that 
 the shorter the wave length, the less does the slit have to be 
 moved out from the central line in order to expose an odd 
 number of elements and so to cause illumination at the centre. 
 Therefore, when white light falls on a slit, the diffracted light 
 of shortest wave length lies nearest the white band which 
 passes straight through the slit. The deviation of the colors 
 is the reverse of that caused by dispersion. The wave length 
 of the violet is the least, and that of the red, the greatest, of 
 the colors which constitute the spectrum. 
 
 It may be as well to mention here, though it was not used 
 by Fresnel, the , instrument called the diffraction grating. It 
 consists of a very large number of equidistant slits. These are 
 made by cutting lines with a diamond upon a glass plate or 
 upon a plate of speculum metal. It may be shown by theory 
 that when light of one wave length falls upon such a grating 
 from a point source, the diffraction bands produced are very 
 intense and narrow. The distance between them depends on 
 the distance between the successive cuts on the grating, and 
 may be made large by making the cuts near together. In the 
 gratings used in most spectroscopic work, there are from 15,000 
 to 20,000 cuts to the inch, and the diffraction bands are so 
 widely separated that only three or four of them can appear
 
 LIQBT. 231 
 
 in front of the grating. When white light falls on such a grat- 
 ing, its constituent colors are diffracted at different angles, 
 according to their wave lengths. With fine gratings, having a 
 large number of lines, the spectrum thus obtained is of great 
 purity. Since the distance between the diffraction bands pro- 
 duced by different wave lengths is proportional to the difference 
 between the wave lengths, this spectrum is called a normal 
 spectrum. 
 
 148. Polarized Light. In the form of the wave theory first 
 used by Fresnel, the waves were thought of as longitudinal. 
 That is, the vibrations of the medium were supposed to take 
 place to and fro in the line of transmission. An example of 
 such a mode of vibration was already known in the case of 
 sound, and it was also known that a fluid, as the ether was 
 then assumed to be, could only vibrate in that manner. When 
 Young and Fresnel tried to apply the wave theory to the ex- 
 planation of double refraction and polarized light, they found 
 it impossible to make any headway so long as they retained 
 the hypothesis of longitudinal vibrations. In fact it is clear 
 that such vibrations cannot present any such distinct differ- 
 ences on different sides of the ray as occur in the case of polar- 
 ized light. Both Young and Fresnel determined, therefore, to 
 abandon the hypothesis of longitudinal vibrations, and to adopt 
 in its stead the hypothesis that the vibrations are more or 
 less transverse to the line of progress. Young did little more 
 than indicate his adoption of this view, and its development 
 was entirely due to Fresnel. 
 
 In order to verify the hypothesis of transverse vibrations, 
 so far as it can be done by experiment, Fresnel and his friend 
 Arago executed a series of experiments on the interference of 
 polarized light. To understand the statement of their results, 
 it must be mentioned that the two polarized rays which pass 
 through a crystal of Iceland spar are said to be polarized in 
 opposite planes, meaning thereby in planes at right angles to 
 each other, but parallel with the direction of the ray. Fresnel 
 and Arago tried Young's experiment to obtain interference, 
 only using polarized light, variously modified, instead of nat-
 
 282 LIGHT. 
 
 ural light. They found that when polarized light was used as 
 the source and was not modified before it fell upon the two 
 openings, the interference phenomena obtained were just the 
 same as those obtained with ordinary light. By interposing 
 properly prepared crystals in the paths of the polarized light 
 falling from the source on the two openings, they polarized the 
 beams which fell on the openings in opposite planes, and then 
 found that interference did not occur. This result is in accord 
 with the hypothesis that the vibrations of light are transverse 
 to the line of progress, and are at right angles to each other in 
 oppositely polarized rays. For, when the rays from the two 
 openings in the first experiment were similarly polarized, they 
 would be in the same plane and so could interfere destructively. 
 When, in the second experiment, the two rays were polarized 
 in opposite planes, the vibrations would be perpendicular to 
 each other, and so could never destroy each other by interfer- 
 ence ; for it is plain that two vibrations at right angles to each 
 other can never act on one particle so as to keep it at rest. 
 Fresnel concluded from these experiments that the hypothesis 
 of transverse vibrations was confirmed. 
 
 It may be well to consider at this point Fresnel's descrip- 
 tion of common light on the hypothesis of transverse vibra- 
 tions. He supposed it to be such a motion in the medium as 
 may be obtained by the superposition of two simple harmonic 
 motions transverse to the line of progress and at right angles 
 to each other. The path of a point describing such a motion 
 is, in general, an ellipse. Two such motions may be superposed 
 so as to produce interference so long as the elliptic paths are 
 similar. The fact that interference can be obtained between 
 two rays of light which differ in length by 2,500,000 wave 
 lengths, shows that the vibrations at the source remain sim- 
 ilar, and send out similar disturbances through space, for a 
 time containing at least 2,500,000 periods of the vibration. On 
 the other hand, it is also evident that the vibration from a 
 source does not remain always the same. If it did do so, it 
 would be polarized, and the two rays into which it is broken 
 by a doubly refracting crystal would differ in intensity from
 
 LIGHT. 233 
 
 each other, and would have different relative intensities for 
 different positions of the crystal. Now observation shows that 
 both the rays transmitted by the crystal have the same inten- 
 sity, when the light used is common light. We can explain 
 this only by supposing that the phase of one of the component 
 vibrations, or the phases of both of themj change abruptly from 
 time to time, so as to alter the polarization of the vibration. 
 Since over 500 million million vibrations are executed by yel- 
 low light in one second, a sufficient number of such changes 
 may occur in that time to give the two component vibrations 
 into which the common light is divided by the crystal equal 
 average intensities, and yet those vibrations may continue in 
 one phase long enough to account for the interference of rays 
 whose paths differ by millions of wave lengths. This descrip- 
 tion of common light is confirmed by another experiment of 
 Fresnel on the interference of polarized light. He found that 
 if light originally polarized before it entered the source was 
 divided into two beams polarized at right angles, which fell on 
 the two openings, these beams, which, in this condition, would 
 not interfere, could be made to interfere, if their planes of 
 polarization were brought into coincidence. On the contrary, 
 if the light which fell on the source was not polarized, and if 
 its two beams which fell on the openings were first polarized 
 in opposite planes, the beams thus formed did not interfere, 
 even when they were brought into the same plane of polariza- 
 tion. In the latter experiment, we notice that the two polar- 
 ized beams which fall on the two openings contain the two 
 components of the elliptic vibration of common light. Accord- 
 ing to our hypothesis, these components change their phases 
 occasionally, and there is no reason to think that their changes 
 of phase will always occur together. If, therefore, those com- 
 ponents of them which fall in the similarly polarized beams 
 upon the receiving screen are at one instant so related as to 
 interfere at one point, they will not remain so long enough for 
 the interference to be perceptible. The interference bands will 
 pass from one point to another on the screen so many times 
 in a second that the illumination of the screen will appear 
 uniform.
 
 234 LIGHT. 
 
 Fresnel employed the hypothesis of transverse vibrations to 
 explain polarization by reflection. It had been discovered a few 
 years before by Malus that when light is incident upon a re- 
 flecting surface of water or glass, the light in both the reflected 
 and refracted beams is generally partially polarized. For a 
 certain angle of incidence, called the polarizing angle, the re- 
 flected light is completely polarized. Most reflecting sub- 
 stances, except the metals, have this property of polarizing 
 light by reflection. It was discovered by Brewster that com- 
 plete polarization occurs when the angle of incidence is such 
 that the reflected ray and the refracted ray are at right angles 
 to each other, or when the tangent of the polarizing angle is 
 equal to the index of refraction. The planes of polarization in 
 the reflected and refracted beams are perpendicular to each 
 other. When the polarization is complete in the reflected beam, 
 the polarization is a maximum in the refracted beam. By 
 making certain suppositions regarding the relations of the 
 components of the vibrations at the reflecting surface, which 
 were not entirely in accord with mechanical principles, 
 although he treated the vibrations as if they were the vibr?- 
 tions of some sort of matter, and as if they conformed to the 
 law of conservation of mechanical energy, Fresnel was able to 
 show that light transmitted by vibrations occurring in the 
 plane of incidence will not be reflected, if the angle of incidence 
 is the polarizing angle, but will be entirely refracted. The 
 effect of reflection at the polarizing angle upon common light 
 is therefore to reflect the component of its elliptic vibration 
 which is perpendicular to the plane of incidence, while the 
 other component, lying in the plane of incidence, is contained 
 only in the refracted beam. At any other angle of incidence 
 than the polarizing angle, both these components occur in both 
 the reflected and the refracted beams, but in different propor- 
 tions, so thf>t those beams are partially polarized. Fresnel's 
 theory led to certain relations between the intensities of the 
 reflected and refracted beams, which he found to be in agree- 
 ment with observation. 
 
 When polarized light was first studied in connection with 
 double refraction, the plane of polarization was specified by
 
 LIGHT. 236 
 
 reference to a definite plane in the crystal. A typical crystal 
 of Iceland spar is a rhombohedron, bounded by six equal faces, 
 each of which is a similar rhombus. The line drawn from one 
 obtuse angle of this rhombohedron to the opposite obtuse angle 
 is called the axis of the crystal. It is the line marking the 
 direction along which no double refraction takes place. We 
 have already called it the optic axis. Any plane perpendicular 
 to one of the faces of the crystal and parallel to the optic axis 
 is called a principal plane. The ordinary ray emerging from 
 the crystal was conventionally said to be polarized in the prin- 
 cipal plane. The extraordinary ray, which is polarized oppo- 
 sitely to the ordinary ray, was then said to be polarized in a 
 plane at right angles to the principal plane. When a polarized 
 ray falls upon a reflecting surface at the polarizing angle and 
 in such a way that its plane of polarization is perpendicular 
 to the plane of incidence, it is not reflected. According to 
 Fresnel's hypothesis, therefore, its vibrations are in the plane 
 of incidence. Fresnel concluded, therefore, that the vibrations 
 of the polarized ray are perpendicular to the plane of polar- 
 ization. 
 
 By modifying one of Fresnel's hypotheses, F. Neumann and 
 McCulIagh developed a theory of polarized light which differed 
 from that of Fresnel in supposing that the vibrations are in 
 the plane of polarization. Both theories were able to account 
 for all known facts of observation. For many years they stood 
 as alternative theories. They have been explained and recon- 
 ciled by the electromagnetic theory of light. 
 
 Fresnel was able to explain the double refraction in Iceland 
 spar and to develop a general theory of double refraction, 
 which applies to all sorts of crystals, on the hypothesis of 
 transverse vibrations. To do this he studied the effect of a 
 disturbance set up inside a body, like a crystal, in which the 
 elasticity is different in different directions. He showed that 
 any general elliptic vibration will be t oncn resolved into 
 two vibrations at right angles to each other, and that these will 
 be transmitted in different directions in the crystal with differ- 
 ent velocities. The surface reached at any instant by the vibra-
 
 236 LIGHT. 
 
 tions which pass out in all directions from the disturbed centre 
 is what is called the wave surface. In most crystals, the wave 
 surface is a complicated one, formed of two sheets, which 
 touch each other at four symmetrically arranged points. These 
 points may be connected in pairs by two lines which pass 
 through the centre. These lines are optic axes, and a ray trav- 
 elling along either one of them will not be doubly refracted. 
 For certain classes of crystals, of which Iceland spar is an 
 example, the wave surface reduces to a sphere and an ellipsoid, 
 which touch each other at the ends of one of the axes of the 
 ellipsoid. The line joining these points of contact is the optic 
 axis. Crystals of this sort are called uniaxial crystals; crys- 
 tals of the other sort are biaxial. 
 
 By the use of the Fresnel wave surface, the directions of 
 the two rays formed by double refraction may be calculated 
 for different angles of incidence, and the theory can be thus 
 tested. The most minute observation has found no point of 
 disagreement between the conclusions of the theory and the 
 results of observation. The form of the wave surface in crys- 
 tals is thus proved to be that developed by Fresnel. This re- 
 sult does not, however, confirm the hypotheses from which 
 Fresnel deduced it. Other forms of the theory lead to the same, 
 or practically to the same, form of the wave surface. 
 
 The hypothesis of transverse vibrations received additional 
 confirmation from its success in explaining certain phenomena 
 discovered by Arago. Arago found that if a parallel beam of 
 polarized light was received upon a crystal so placed as to 
 extinguish the beam, the light could be made to reappear by 
 interposing in the path of the beam a thin sheet of mica. The 
 light which thus appeared was colored, and the color changed 
 as the crystal through which it was observed was rotated. 
 To show this experiment we use two crystals of Iceland spar, 
 called Nicol's prisms. A Nicol's prism is a long prism of Ice- 
 land spar which has been cut through diagonally from one 
 obtuse angle to the other, and cemented together again with 
 Canada balsam after the new faces have been polished. A 
 beam of light which falls on one end of this crystal is divided
 
 LIGHT. 237 
 
 into two oppositely polarized beams, which proceed through 
 the crystal till they meet the surface of the Canada balsam. 
 If the crystal has been divided in the proper way, the ordi- 
 nary ray will meet the balsam at an angle greater than its 
 critical angle. It will consequently be totally reflected and 
 will not emerge from the other end of the crystal. The ex- 
 traordinary ray, on the contrary, meets the balsam within its 
 critical angle, and is therefore partly transmitted through it, 
 and emerges from the crystal. The Nicol's prism may be thus 
 used to obtain a beam of polarized light. In trying Arago's 
 experiment, one Nicol's prism is used as the polarizer. The 
 other, called the analyser, is placed in the path of the polar- 
 ized beam and the light which comes through it is observed 
 either on a screen or by the eye. When the analyser is 
 turned so that its principal plane is at right angles to that 
 of the polarizer, no light passes through it. If we now inter- 
 pose between the two prisms a sheet of mica, colored light 
 passes through the analyser. In only two positions of the 
 mica sheet will this not occur, when its principal plane is 
 either parallel with, or perpendicular to, the plane of polar- 
 ization of the incident beam. For other positions of the mica, 
 no position of the analyser can be found in which light will 
 not pass through it. Fresnel explained this phenomenon in the 
 following way: The vibrations of the polarized beam which 
 loll on the mica are resolved into two components perpendicu- 
 lar to each other. The velocities of these components are dif- 
 ferent, and when they emerge from the mica one of them has 
 gained a fraction of a wave length on the other, so that they are 
 then in different phases. Because the wave lengths of different 
 colors are different, the differences in ph"ase between the two 
 emergent beams are different for the different colors. After 
 emergence, these beams proceed to the analyser, which resolves 
 each of them into two perpendicular components, and transmits 
 one component of each. The components transmitted are par- 
 allel with each other, and may therefore interfere. If they 
 happen to be in opposite phases, they destroy each other. If 
 they are in the same phase they enhance each other. For a
 
 238 LIGHT. 
 
 given thickness of the mica some of the constituents of the 
 original white beam will thus be destroyed, and the light which 
 will pass will appear colored. The light which is rejected by 
 the analyser contains those colors which are absent in the 
 transmitted light, so that if the analyser is turned so that it 
 transmits the light which it formerly rejected, the comple- 
 mentary color appears. 
 
 Similar effects may be produced by the use of thin sheets of 
 crystals other than mica. 
 
 By using divergent or convergent light instead of parallel 
 light, very complicated systems of colored figures may be pro- 
 duced. The peculiarities of these systems may be calculated 
 from a knowledge of the optical properties of the particular 
 crystal which is interposed in the beam. When the results of 
 calculation are compared with the results of observation, a very 
 complete agreement is found between them. This general result 
 furnishes additional confirmation of the Fresnel wave surface. 
 
 Arago discovered a peculiar effect produced by quartz on 
 polarized light, which was also explained by Fresnel. Quartz 
 is a uniaxial crystal, and light falling perpendicularly upon a 
 plate of quartz cut perpendicularly to the optic axis is not 
 doubly refracted. If, however, an analyser is so placed in a 
 polarized beam as to extinguish it, and if then such a plate of 
 quartz is interposed in the polarized beam, light will again 
 come through the analyser. Unless the plate of quartz is too 
 thick, this light is colored. It is not altered by rotation of the 
 quartz around the beam as an axis, but when the analyser is 
 rotated, the color changes continually. If light of one color is 
 used in this experiment, it will pass through the analyser when 
 the quartz plate is introduced, and if the analyser is then 
 turned through a certain angle, which is different for each 
 color, the light will be extinguished. From this experiment it 
 is easy to see that the different colors which appear, when 
 white light is used, are due to the suppression of some of the 
 constituents of white light in each position of the analyser. 
 This action of quartz is known as the rotation of the plane of 
 polarization. Many other substances, among them solutions of
 
 sugar and of other organic bodies, were subsequently found to 
 possess the same property. From the fact that it is possessed 
 by solutions, in \vhich it is impossible to suppose that the act- 
 ive molecules have any definite directions, such as they may be 
 supposed to have in crystals, we conclude that this action on 
 polarized light is due to the structure of the molecule itself, 
 or perhaps of one of its atoms. Fresnel explained this phe- 
 nomenon by supposing that the polarized ray, on entering the 
 quartz, sets up two circular vibrations in opposite senses. It 
 is easy to see that the resultant of two such vibrations will be 
 a rectilinear vibration like that of the incident ray. He further 
 supposed that the velocity in the crystal of one of these vibra- 
 tions is greater than that of the other. If this is the case, the 
 two circular vibrations, on their emergence from the quartz, 
 will combine again to produce a rectilinear vibration, which is 
 inclined to the one which entered the quartz by an amount 
 proportional to the thickness of the plate. If the circular 
 vibration which is traveling faster is the one in which the 
 rotation is clock-wise to an observer looking along the ray, the 
 plane of polarization of the emergent light will be turned clock- 
 wise. If the other circular vihrnti"ii travels faster through the 
 quartz, the plane of polarization will be turned counter-clock- 
 wise. Specimens of quartz are found which show each of these 
 rotations. They are called right-handed and left-handed, 
 respectively. By an ingenious combination of quartz prisms 
 properly cut, Fresnel was able to separate the two circularly 
 polarized rays assumed by him in this explanation, and so to 
 prove its correctness. 
 
 In his study of the reflection of polarized light Fresnel was 
 led to consider the effect of total reflection upon the plane of 
 polarization. In that case he found that when the vibration in 
 the polarized beam is not at right angles to the plane of inci- 
 dence, that component of it which lies in the plane of incidence 
 has its phase reversed by the reflection. In general the com- 
 bination of the new component thus produced with the other 
 component which is reflected without change produces an ellip- 
 tic vibration. By combining two such total reflections, taking
 
 240 LIGHT. 
 
 place at the proper angles, the vibration of the reflected beam 
 becomes circular. The beam is then said to be circularly polar- 
 ized. Circular polarization may also be produced by the use 
 Of a sheet of mica, whose thickness is such that one of the two 
 rays formed in it by double refraction gains a quarter of a 
 wave length on the other in its passage through the sheet. This 
 sheet is placed in the path of a polarized beam in such a posi- 
 tion that its principal plane makes an angle of 45 with the 
 plane of polarization. In this case the two components which 
 emerge from it contain vibrations of equal magnitude and dif- 
 fering in phase by one quarter of a period. These two vibra- 
 tions combine to produce a circular vibration. Such a sheet 
 of mica is called a quarter-wave plate. 
 
 When polarized light is incident obliquely upon a polished 
 metallic surface, the reflected light is elliptically polarized. 
 
 It was discovered by Faraday that when a beam of plane 
 polarized light traverses a magnetic field in the direction of 
 its lines of force, the plane of polarization is rotated. The 
 amount of rotation depends upon the strength of the field and 
 upon the substance through which the light is passing. It was 
 found by Faraday first in a glass of a peculiar composition. 
 The rotation produced by the magnetic field differs from that 
 produced by quartz in one important respect. If the beam 
 which has been rotated by the magnetic field is received on a 
 plane mirror and sent back through the field again, it under- 
 goes an additional rotation in the same sense. On the other 
 hand, a beam which is sent back through the quartz undergoes 
 rotation in the opposite sense, so that its plane of polarization 
 becomes the same as that of the incident beam. Magnetic rota- 
 tion of the plane of polarization is explained in the same gen- 
 eral way as the rotation by quartz. The two circularly polar- 
 ized beams have recently been separated by Brace. 
 
 149. Spectrum Analysis. When salts of the elements are 
 introduced into a flame, they give the flame characteristic 
 colors. For example, sodium chloride or sodium carbonate will 
 color the flame yellow. If the light from this flame passes 
 through a narrow slit and is received on a prism so as to pro-
 
 LIGHT. 241 
 
 duce a pure spectrum, certain parts of the spectrum are found 
 to be much more intense than the rest of it. In the case of the 
 sodium salts a narrow band or line, very brilliantly illumin- 
 ated, appears in the yellow. This yellow line of sodium was 
 noticed by Herschel. By the investigations of Bunsen and 
 Kirchhoff, 1859-1862, it was shown to be characteristic of the 
 presence of sodium vapor in the flame, so that whenever this 
 line can be detected, it may be inferred that sodium is present. 
 Bunsen studied the characteristic lines, or, as we may say, the 
 spectra of different elements, and showed that they can be used 
 as a means of detecting the presence of those elements in the 
 substance by which the flame is colored. Elements which can 
 not be vaporized in a flame may be vaporized by the heat of the 
 electric arc, and will then give similar characteristic line spec- 
 tra. When the electric spark is passed through an elementary 
 gas, a similar characteristic spectrum is produced. 
 
 This method of detecting the presence of a particular ele- 
 ment in a compound by means of its characteristic spectrum, 
 is called spectrum analysis. 
 
 If the spectrum of a gas is observed as the pressure upon 
 it is increased, it is found that the lines, which at first are 
 sharp and narrow, gradually broaden out into bands, and as 
 the pressure is still further increased, these bands overlap, 
 until the spectrum becomes continuous, like that of an incan- 
 descent solid. 
 
 150. Fraunhofer's Lines. When a pure spectrum is formed 
 with sunlight, it is found to be crossed by a great number of 
 dark lines. Some of the most intense of these lines were ob- 
 served by Wollaston, and they were more accurately studied by 
 Fraunhofer. The light from an ordinary flame or from an 
 incandescent body does not show these lines. 
 
 It was noticed by several observers that one of the most 
 conspicuous of the Fraunhofer lines, in the yellow light of the 
 spectrum, coincided in position with the yellow line of the 
 spectrum of sodium. This coincidence was verified by the exact 
 observations of Kirchhoff, who was working with Bunsen on 
 the development of spectrum analysis. Kirchhoff suspected
 
 242 LIGHT. 
 
 that the dark line in the solar spectrum is due to the absorp- 
 tion of light, coming from the central part of the sun, by the 
 vapor of sodium in its outer atmosphere. To show that this 
 may be the case, he observed the spectrum of the white light 
 coming from incandescent lime, heated by the oxyhydrogen 
 flame, when a flame containing sodium vapor was placed in 
 front of the slit. He found, with those conditions, that a dark 
 line appeared in the spectrum, coinciding in position with the 
 bright yellow line which the sodium vapor itself would have 
 given. He explained the production of this dark line by the aid 
 of the principle of resonance. From the fact that sodium vapor 
 emits light of a certain period, as shown by its giving rise to 
 the characteristic yellow line of its spectrum, it is evident that 
 those elements of sodium vapor which emit light execute vibra- 
 tions of that period. Now, if light of all periods of vibration 
 falls upon the sodium vapor, those vibrations which are not 
 similar to the natural vibrations of the sodium will pass on 
 without disturbing it, but those vibrations which have the same 
 period as that of the sodium will increase its natural vibration 
 by giving to it impulses properly timed, and so will them- 
 selves be diminished in intensity, or absorbed. The dark line 
 which is thus cast in.the spectrum is not black, but simply not 
 so intensely illuminated as the regions on either side of it. 
 
 A comparison of the Fraunhofer lines of the solar spectrum 
 with the spectra of the various elements led to the discovery of 
 many coincidences similar to that described in the case of 
 sodium. In some cases, as in that of iron, for example, these 
 coincidences extend to hundreds of lines. In every case in 
 which such coincidences can be proved, we infer the presence 
 of the particular element giving the spectrum in the outer 
 fitmosphere of the sun. 
 
 When a colored transparent body is placed in the path of a 
 beam which forms a spectrum, there often appear particular 
 portions of the spectrum in which the light is less intense than 
 it was before. These darker regions are called absorption 
 bands. They are characteristic of the particular substance 
 which intercepts the light, and may be used as a means of 
 analysis.
 
 LIGHT. 243 
 
 Kirchhoff found by a general theoretical investigation that 
 the absorption of yellow light by sodium vapor is an example 
 of a perfectly general law, that any body will absorb those 
 waves of light which it will itself emit when self-luminous. 
 The ratio between the emissive power and the absorptive power 
 is the same for all substances at the same temperature. Those 
 bodies which absorb all colors, also emit all colors when they 
 become self-luminous. Those bodies which absorb only partic- 
 ular colors, emit only those particular colors. 
 
 We are now in a position to consider the question of the 
 color of bodies. All bodies which are not self-luminous are 
 seen by means of the light reflected by them, coming from the 
 sun, or from some other self-luminous source. Very many 
 bodies show the same color when examined by transmitted 
 light as when examined by reflected light. To explain their 
 color, we suppose that some of the constituents of the white 
 light which falls upon them are absorbed, and that the light 
 which is reflected to the eye, and by which the body is seen, has 
 penetrated sufficiently within the body to allow this absorption 
 to deprive it of those constituents. We therefore see it really 
 by light which has traversed enough of thce body for absorption 
 to have its full effect. 
 
 There are many other bodies, however, for which the color 
 in the transmitted light is different from that in the reflected 
 light. With them the reflection seems to occur at the surface. 
 It is found in every such case that absorption bands appear in 
 the transmitted light, and that the light which is reflected is 
 exactly that which is wanting in the transmitted beam. These 
 bodies show also another peculiarity, which was discovered by 
 Christiansen in fuchsine, and which was fully studied by 
 Kundt. This peculiarity i? oallod anomalous dispersion. In 
 very many cases, the spectrum formed by a prism of a particu- 
 lar substance, such as glass, for example, has the colors 
 arranged in the order of their wave lengths. A dispersion of 
 this sort is supposed to be according to law, and any deviation 
 from it is anomalous. When light is sent through a prism of 
 the substance showing anomalous dispersion, the order of the
 
 244 LIGHT. 
 
 colors of the spectrum is not the order of the wave lengths. It 
 was found by Kundt to be a general law that the colors which 
 are displaced from their position in the ordinary spectrum are 
 those which lie on either side of an absorption band. The color 
 whose wave length is longer is displaced toward the violet end 
 of the spectrum, and that whose wave length is shorter, toward 
 the red end. 
 
 All these peculiarities of substances which show surface 
 color and anomalous dispersion can be explained by supposing 
 that the elements of their structure which can emit light have 
 vibrations of their own of the same period as the light which 
 the substance absorbs. 
 
 151. The Extent of the Spectrum. Herschel examined the 
 heating effect produced by the different parts of the spectrum 
 by placing a thermometer in it; and found that as the bulb of 
 the thermometer was moved down toward the red end of the 
 spectrum the heating effect became more and more pronounced, 
 and that a still greater heating effect appeared when the bulb 
 was placed in the region just beyond the red end of the spec- 
 trum. He concluded that rays exist of longer wave length than 
 the red rays. When the action of light on salts of silver was 
 discovered, on which photographic processes depend, it was 
 found that the most active region in this respect lay outside the 
 violet end of the spectrum. The invisible rays which produced 
 the heating effect were at first called heat rays, and those which 
 produced the chemical effect, actinic rays; but it is easily seen 
 that there is no reason for considering them to be essentially 
 different from the visible rays. Accordingly it has been found 
 that all these rays, withoxit exception, produce a heating effect, 
 and that the chemical effect has been caused by so many of 
 them, that the conclusion is fully warranted that the rays 
 from a luminous body or from any body are of the same nature, 
 and differ only in- the lengths of the waves whose direction of 
 transmission they indicate. The examination of the radiation 
 from incandescent vapors shows that many spectral lines are 
 emitted by them which lie in the invisible parts of the spec- 
 trum. And similarly, multitudes of Fraunhofer lines are de- 
 tected in the invisible parts of the solar spectrum.
 
 LIGHT. 246 
 
 The shortest waves in the extreme violet are about 0.0004 
 millimetres long; the longest in the extreme red, about 0.0007 
 millimetres long. Photographic methods have detected waves 
 0.0001 millimetres long, and observations of the heating effect 
 have detected waves 0.02 millimetres long. 
 
 Balmer showed that the spectral lines of hydrogen are so 
 disposed that their wave lengths, or the number of vibrations 
 corresponding, can be calculated by the aid of a general form- 
 ula. In applying this formula each spectrum line is assigned 
 one of the natural numbers, according to its position in the 
 hydrogen spectrum, beginning with 3, assigned to the line corre- 
 sponding to the longest wave length, and the insertion in the 
 formula of the number assigned ip the line leads to the number 
 of vibrations corresponding to the line. Similar series, calcu- 
 lable by similar, though more complicated formulae, have been 
 discovered in the spectra of many other elements. 
 
 152. Fluorescence. Certain substances, such as fluor-spar 
 or solutions of chlorophyll or of sulphate of quinine, when 
 placed so as to receive a narrow beam of light, become self- 
 luminous in a peculiar way. The light emitted by them seems 
 to originate in the path of the beam, and has a characteristic 
 color and spectrum, depending on the nature of the substance. 
 The phenomenon is known as fluorescence. According to Stokes 
 the wave length of the emitted light is always less than that of 
 the light which excites it. This law has recently been dis- 
 proved by Nichols and Merritt. The production of the light 
 may be explained as a species of resonance. 
 
 By rapidly cutting off and renewing the incident beam, it 
 has been shown that the emitted light persists, though often for 
 only a short time, after the incident beam is cut off. Fluor- 
 escence is therefore apparently not distinct from phospho- 
 rescence, that is, from the phenomenon of persistent lumin- 
 escence excited by exposure to light in certain substances, such 
 as sulphide of calcium. 
 
 153. Zeeman Effect. It has recently been discovered by 
 Zeeman that the vibrations of a vapor which emits light are 
 peculiarly modified if they are executed in a magnetic field.
 
 24G LIGHT. 
 
 We need not describe this modification further than to say, 
 that a single spectral line which the vapor would ordinarily 
 emit is divided, when the vapor is in a magnetic field, into two 
 or more lines, and that the light in these lines is differently 
 polarized. 
 
 We are not able to explain the Zeeman effect, nor indeed to 
 give an adequate explanation of the hypotheses upon which our 
 whole theory of light has been based, by any purely mechani- 
 cal theory of the ether and of the nature of light. It is now 
 quite certain that what we have called the vibrations of light 
 are periodic electric disturbances in the ether, and that the 
 various modifications impressed upon them by material bodies 
 are due to the electric relations of those bodies. W T e shall dis- 
 cuss the electromagnetic theory of light in connection with our 
 study of electricity. 
 
 154. Velocity of Light. The first direct determination of 
 the velocity of light, by a method which did not involve astro- 
 nomical measurements, was made by Fizeau in 1849. He used 
 a toothed wheel which could be rapidly revolved at a known 
 rate. When the wheel was still, a beam of light was sent out 
 through one of the gaps between the teeth, was received on a 
 mirror set up at a great distance, and reflected back through 
 the same gap. The wheel was then set in rotation. At first, 
 as it turned slowly, the light sent back from the mirror still 
 passed through the same gap, but as the rotation was increased, 
 the neighboring tooth moved forward into the path of the 
 light, so as to intercept it. When the rotation of the wheel was 
 doubled, the returning light passed through the next gap. In 
 this way, by increasing the rate of rotation, the returning light 
 was alternately intercepted and transmitted. By observing the 
 different rates of rotation, it was easy to determine the time 
 occupied by the light in passing from the wheel to the mirror 
 and back again to the wheel. The velocity of light determined 
 in this way was found to be about 315 million metres per 
 second. 
 
 Another method of determining the velocity of light was 
 employed by Foucault, in 1862. He used for that purpose a
 
 LIGHT. 247 
 
 small mirror, which could be rotated rapidly around an axis 
 parallel to its face. When the mirror was at rest, light was 
 allowed to fall on it from a slit, and was reflected by it to a 
 fixed mirror. This mirror sent the light back over the same 
 path, so that it fell on the slit. When the mirror was rotated 
 the angle at which it received the returning light was different 
 from that at which it had sent out the same light from the 
 slit, and the returning light reflected from it no longer fell on 
 the slit. The velocity of light was determined by measuring the 
 displacement of the image of the slit sent back from the revolv- 
 ing mirror and the rate of rotation of the mirror. Foucault 
 obtained a result for the velocity of light which was a little 
 smaller than that obtained by Fizeau. This method has been 
 used by Michelson and by Newcomb to obtain the velocity of 
 light which is now considered the standard. Newcomb gives it 
 as 299,860 kilometres per second. For ordinary purposes we 
 may take it as equal to 300 million metres per second. 
 
 It was by the use of this method that Foucault proved, in 
 1850, that the velocity of light in water is less than in air. 
 
 155. Effect on the Velocity of Light of the Motion of 
 Bodies. When the wave theory of light was first studied by 
 Fresnel and Arago, the question of the aberration of light had 
 to be considered." Arago perceived that, on the accepted expla- 
 nation of aberration, the aberration of a star ought to be dif- 
 ferent from that ordinarily obtained for it 3 if the tube of the 
 telescope by which it was observed was filled with water. Ob- 
 servation showed, however, that the aberration was the same 
 when thus determined as it had previously been found to be. It 
 was shown by Fresnel that this result could be explained if the 
 velocity of light was supposed to be affected by the velocity of 
 the body through which it was passing, according to a certain 
 law. In order to test this hypothesis, Fizeau observed the 
 change produced in the velocity of light by sending it through 
 a stream of water. The results of his observations, which have 
 since been confirmed by Michelson and Morley, were in agree- 
 ment with Fresnel's formula.
 
 248 LIGHT. 
 
 These experiments are consistent with the hypothesis that the 
 ether is at rest, and does not share in the motion of bodies 
 moving through it. On the other hand, Michelson and Morley, by 
 the use of an instrument called the interferometer, compared the 
 velocity of light in the direction of the earth's motion with that of 
 light perpendicular to the earth's motion, and found that they 
 could not detect any difference, such as would be expected if the 
 ether is at rest. This result is at variance with most of the other 
 facts known bearing on the subject, and while it is accepted as 
 proved, the conclusion that the ether moves with the earth is not 
 generally drawn. 
 
 The question of the action of moving matter upon the ether, 
 and of the way in which the motion of matter affects the prog- 
 ress of light through it has not yet been solved. Such explana- 
 tions as have been given depend upon certain assumed electric 
 properties of the ether and of matter, and will be considered in 
 connection with our study of electricity.
 
 MAGNETISM. 249 
 
 MAGNETISM. 
 
 156. The Lodestone. From the earliest antiquity it has 
 been known that certain stones, or pieces of mineral, exist 
 which possess the peculiar property of attracting iron. These 
 stones are called magnets or lodestones. In view of the fact 
 that pieces of iron or steel artificially prepared, and possessing 
 this same property, are also called magnets, it may be best 
 always to designate these natural magnets as lodestones. They 
 contain principally magnetic oxide of iron. 
 
 The attractive power of the lodestone for iron is shown 
 more strongly at some parts of it than at others. By careful 
 selection, lodestones may be found, in which the regions of 
 strongest attraction are two in number, but none are ever found 
 with only one such region. 
 
 It was found by the early observers that when a small piece 
 of iron, such as an iron finger ring, was attracted by the lode- 
 stone, it also acquired the property of attracting iron. The 
 iron thus attracted by it acquired in its turn the same property 
 of attraction. The attractive force developed in each successive 
 piece of iron decreased in intensity as the iron was further 
 removed from the original lodestone. These pieces of iron are 
 said to be magnetized by induction. Later observations showed 
 that iron can be magnetized by induction when it is brought 
 near the lodestone, or near another magnet, without being in 
 contact with it. 
 
 When a lodestone exhibiting two centres of attraction was 
 placed in a light vessel, and floated on the surface of water, it 
 always turned about so that one of the two centres pointed 
 toward the north, and the other toward the south. This prop- 
 erty of assuming a definite direction is generally better exhib- 
 ited by artificial magnets, in which the two centres of attrac- 
 tion are more precisely developed, than it is by the lodestone. 
 The compass is a magnet so arranged that it can turn freely in 
 a horizontal plane and indicate by the direction in which it 
 points the north and south line.
 
 250 MAGNETISM. 
 
 The first scientific study of magnets was made by Gilbert, 
 who published the results of his study in 1600. He found that 
 when the ends of two lodestones of the simplest type were 
 brought near each other, they were attracted to each other if 
 one of them was an end which would point toward the north, 
 and the other an end which would point toward the south. If 
 ends which would point toward the north, or ends which would 
 point toward the south, were brought near each other, they 
 repelled each other. Gilbert called these ends, at which the 
 forces of attraction and repulsion were most strongly exhib- 
 ited, poles, the one which pointed toward the north being called 
 the north pole, the other, the south pole. We may express Gil- 
 bert's discovery by saying that a north pole of a magnet will 
 attract a south pole of another magnet, and that the north 
 poles or the south poles of two magnets will repel each other. 
 
 Gilbert conceived of the magnetic condition of a lodestone, 
 or of any magnet, as due to some sort of arrangement through- 
 out its whole body. The experiment by which he was led to 
 this conclusion consisted in cutting a lodestone exhibiting two 
 poles into two parts, by a section across the line joining the 
 poles. When the two parts thus made were separated and 
 examined, it was found that the poles which they originally 
 possessed had not been altered, but that two new poles had been 
 developed, one in each piece, of a sort different from that 
 already in it. Each piece was therefore a complete magnet, 
 having north and south poles. When the newly made north 
 and south poles were brought near each other, they attracted 
 each other, and when the two pieces were allowed to meet, the 
 new poles disappeared, and only the original poles remained. 
 It has been shown that whenever a piece of any size is cut off 
 from a magnet, it is always itself a complete magnet, and it is 
 inferred that if the molecules of a magnet could be separated 
 from each other, each of them would be a complete magnet. 
 
 157. Artificial Magnets. As has already been described, a 
 piece of iron may be made a magnet by bringing it near a lode- 
 stone. If it is removed from the lodestone and tested, it will be 
 found to be a magnet still. The slightest disturbance of it, by
 
 
 MAONKTI>M. 251 
 
 striking or jarring it, will cause it to lose its magnetic condi- 
 tion. If the iron is in the form of steel, however, it will retain 
 the magnetic condition induced in it by the lodestone to a very 
 great degree. A piece of steel thus prepared is called an arti- 
 ficial magnet. Since its magnetic condition is retained by it 
 indefinitely, in ordinary circumstances, it is also called a per- 
 manent magnet. It is with such steel magnets that we can 
 most easily carry out the experiments which Gilbert described 
 as carried out by him with the lodestone. 
 
 The circumstances involved in magnetization by induction 
 should be more fully examined at this point. To do so we 
 present one end of a soft iron rod to the north pole of a perma- 
 nent magnet, and examine the magnetic condition of the rod. 
 It is found that the rod is magnetized in such a way that the 
 end near the north pole of the magnet is a south pole, and the 
 other end is a north pole. If one end of the rod is presented to 
 the south pole of the magnet, that end becomes a north pole, 
 and the more distant end a south pole. A similar result is 
 obtained if one end of a row of short iron rods is presented to 
 the pole of the magnet. Each of the rods then becomes a mag- 
 net, with a pole different from that of the inducing magnet in 
 the end nearer it, and a pole like that of the inducing magnet 
 in the more distant end. We shall subsequently describe the 
 region around a magnet, in which magnetic force can be per- 
 ceived by the aid of another magnet, as a magnetic field, and 
 we shall then describe magnetic induction in a somewhat more 
 general way. For the present it is sufficient to say, that a 
 magnetic pole will induce a pole unlike itself in that part of a 
 piece of iron which is nearest to it. The attraction between 
 these two unlike poles draws the iron and the magnet together. 
 It was this attraction which was first observed, and which was 
 for a long time supposed to be the fundamental property of a 
 magnet. It was not until Gilbert examined the action of one 
 magnet on another, and observed the magnetization by induc- 
 tion, that the true relations of a magnet to iron could be under- 
 stood.
 
 252 MAGNETISM. 
 
 A piece of steel may be made a magnet by bringing it in 
 contact with a lodestone or a pole of another magnet. Various 
 operations, by which this contact is made according to certain 
 rules, have been devised, in order to effect this magnetization as 
 uniformly and as powerfully as possible. These methods have 
 all been superseded by a method which depends upon the fact 
 that an electric current sets up a magnetic field around itself. 
 The wire carrying the current is wound into a spiral coil, and 
 the steel bar, which is to be magnetized, is placed in the axis 
 of the coil. The magnets produced in this way are in every 
 way better than those produced by the older methods of con- 
 tact. 
 
 If two similar bar magnets are placed side by side with 
 their like poles contiguous, they produce a magnet which is 
 more powerful than either of them. By larger combinations of 
 magnets, very powerful permanent magnets can be produced. 
 
 158. The Earth as a Magnet. Attention has already been 
 called to the fact that a magnet which is free to turn in a hori- 
 zontal plane will point with one of its ends toward the north. 
 The vertical plane containing the direction in which it points 
 at any place is called the magnetic meridian for that place. If 
 the magnet is mounted on a horizontal axis so that it can turn 
 freely in the magnetic meridian, its north pole, in the northern 
 hemisphere, will generally point downward, making a certain 
 angle with the horizontal w r hich is called the magnetic dip for 
 the place. In the southern hemisphere the north pole points 
 above the horizon. By magnetizing a globe of iron and exam- 
 ining the behavior of small magnets, suspended freely at differ- 
 ent points on its surface, Gilbert reproduced on a small scale 
 the phenomena exhibited by free magnets with respect to the 
 earth. He thus concluded that the earth itself is a magnet, or 
 at least has magnetic poles like those of a magnet, the south 
 magnetic pole of the earth, toward which the north pole of the 
 suspended magnet points, being situated in the northern hemi- 
 sphere. 
 
 This conclusion of Gilbert's is in general correct, but there 
 is still much that is not understood about the magnetic condi-
 
 MAGNETISM. 253 
 
 tion of the earth. The magnetic meridians do not coincide with 
 the meridians of longitude, nor do the lines of equal dip coin- 
 cide with the parallels of latitude. The magnetic equator, at 
 which there is no dip, is an irregular line crossing the geo- 
 graphical equator in two points. The angle between the mag- 
 netic meridian and the geographical meridian at any place is 
 called the magnetic variation or declination at that place. This 
 variation not only differs in different places, according to no 
 well defined law, but it also changes with lapse of time. It 
 never exceeds a certain limit, however, and appears after many 
 years to go through a cycle of values. The variation is also 
 subject to a small diurnal change. The dip at any place is also 
 subject to similar changes. 
 
 159. Law of Magnetic Force. Hitherto we have described 
 in general terms the way in which one magnet acts on another. 
 In order to construct instruments for use in magnetic measure- 
 ments, and in order to advance the theory of magnetism, experi- 
 ments were carried out by Coulomb to determine the law of the 
 force between two magnetic poles; that is, to determine the 
 way in which the force between the poles depends on the poles 
 themselves and on the distance between them. In these experi- 
 ments Coulomb used the instrument called the torsion balance. 
 This instrument depends on the use of a twisted wire for meas- 
 uring the couple by which it is twisted, and is similar in all 
 essential features to that used by Cavendish in determining the 
 gravitation constant. 
 
 With this torsion balance Coulomb proved, first, that the 
 magnetic forces exerted upon the magnet by the earth are 
 always applied at the same two points in the magnet, which 
 points are the north and south poles. The line joining these 
 points is called the magnetic axis, and when the magnet is free 
 to turn, the magnetic axis indicates the magnetic north and 
 south line. The forces acting on these poles, because of the 
 earth's magnetic condition, are equal, parallel, and oppositely 
 directed, so that when the magnet is not in the magnetic merid- 
 ian, it is acted on by a couple which tends to turn it into that 
 meridian. In observations of the action of one magnet on
 
 254 MAGNETISM. 
 
 another, this couple due to the earth must be determined and 
 allowed for. When Coulomb determined the force exerted be- 
 tween two magnet poles at different distances from each other, 
 he found that these forces were to each other inversely as the 
 squares of the respective distances between the poles. This 
 conclusion that the force between two magnet poles varies in- 
 versely with the square of the distance between them, was 
 afterwards confirmed by Gauss in another way. Gauss assumed 
 this law of inverse squares, and calculated by means of it the 
 forces which one magnet will exert upon another when they are 
 placed in certain different positions relative to each other. He 
 then examined by experiment the actual forces exerted in these 
 different positions, and found them to be in accord with his 
 calculations. From this result he inferred the truth of the law 
 upon which the calculations were based. 
 
 160. Quantity of Magnetism. Strength of Pole. Coulomb's 
 law of the force between two magnetic poles recalls the law of 
 gravitation, which also varies inversely with the square of the 
 distance. The law of gravitation, however, contains another 
 statement, namely, that the force is proportional to the inter- 
 acting masses. It is plain that in some way the force between 
 two magnetic poles depends on what we may call the strengths 
 of those poles, and we must now inquire how the strength of a 
 pole can be measured, and how the force between two poles 
 depends on their strength. 
 
 In the case of gravitation, the masses which attract each 
 other are already measured in terms of a unit mass, and, in 
 establishing the law, a direct proof was given that the attrac- 
 tion between two masses is proportional to them both. In the 
 case of magnetism, we have no such independent standard of 
 the strength of a pole, or of what we may call, by analogy, the 
 quantity of magnetism at the pole. We have no evidence that 
 there is such a thing as magnetism. We only know that a 
 piece of steel is in a condition in which it exerts peculiar forces 
 on other similar pieces of steel, and we have no way of meas- 
 uring that condition except by those forces. We are therefore 
 foiced to measure the strength of a pole by the force which it
 
 MAGNKT1SM. 265 
 
 will exert on a standard pole according to some assumed law. 
 The law which we assume is the simplest possible, namely, 
 that the strengths of different poles are to each other as the 
 forces which they will exert on the same pole, if placed at equal 
 distances from it. When the strength of pole is thus defined, 
 the law of magnetic force may be enlarged into the statement 
 that the force between two poles is proportional to the product 
 of their strengths and inversely proportional -to the square of 
 the distance between them. Owing to the fact that the force 
 between two poles depends on the medium which surrounds 
 them, we assume in this definition that the poles are in a 
 vacuum. The force measured in air is not perceptibly different 
 from that in vacuum. 
 
 Our definition of strength of pole has merely compared one 
 pole with another. It is, however, important, and even neces- 
 sary for magnetic measurements, to adopt a unit magnetic 
 pole, in terms of which all other poles may be measured. The 
 way in which such a pole can be defined is best seen by exam- 
 ining the formula which expresses the force between two poles. 
 By introducing a factor of proportion we may write the equa- 
 
 poles m and m', when the distance between them is r. The sym- 
 bol k is the factor of proportion. Now, in the case of the sim- 
 ilar formula expressing the law of gravitation, the two masses, 
 which correspond to the two strengths of pole, are supposed 
 to be known in terms of the unit of mass, and the force and 
 the distance are measured, so that by the substitution of these 
 known quantities in the formula, the constant k is determined 
 as the gravitation constant. In the case of magnetism, it is 
 conceivable that an arbitrarily chosen magnetic pole might be 
 taken as a standard, and other magnetic poles always ex- 
 pressed in terms of it. If this were done, the quantities in the 
 formula would all be known, except the factor k, and so fc 
 would be determined as the constant of magnetic force. It is, 
 however, practically impossible to construct a magnet of such 
 a sort that its pole can conveniently be used as a standard, and
 
 256 MAGNETISM. 
 
 it is equally impossible to preserve it so that the strength of 
 its poles will not change with use and with lapse of time. We 
 therefore proceed otherwise in determining the unit pole. In- 
 stead of arbitrarily choosing a unit pole and then, by observa- 
 tion of the force between two measured poles, determining the 
 value of the factor of proportion, we arbitrarily choose a value 
 of that factor, and then, by observation of the force between 
 two poles, determine their values, or rather the value of their 
 product. To make this choice as simple as possible, we set 
 k=l. On this assumption, the measured force between two 
 poles, at a known distance apart, will enable us to determine 
 the product of their strengths. If the poles are alike, we may 
 thus determine the strength of either one of them. In partic- 
 ular we may define the unit pole, or pole of unit strength, as 
 that pole which will repel another equal and similar pole at 
 unit distance with unit force. In the C. G. S. system, the unit 
 magnetic pole is one which will repel an equal and similar 
 pole, at the distance of one centimetre, with the force of one 
 dyne. 
 
 161. Magnetic Field. When a small magnet is brought 
 near another magnet, it "will be acted on by magnetic forces, 
 and will indicate this action by assuming some definite posi- 
 tion. If we imagine it possible to isolate one of the poles of 
 this magnet, or what amounts to the same thing, if we exam- 
 ine the force exerted on it, we shall find that, in these circum- 
 stances, it is acted on by a force of definite amount and direc- 
 tion. By using such an isolated pole, the forces in the region 
 around a magnet, or around any combination of magnets, can 
 be mapped out with their magnitudes and directions. Such a 
 region, in which a magnetic pole will be acted on by magnetic 
 force, is called a magnetic field. 
 
 The existence of a magnetic field may often be detected 
 when there is no' evidence of the presence of a magnet to which 
 the field can be assigned. Thus there is a magnetic field 
 around a wire carrying an electric current. The magnetic field 
 of the earth is another example.
 
 MAONBTISM. 25.7 
 
 if we lay >a sheet of stiff paper over a strong bar magnet, 
 and scatter iron filings over it, the filings will be arranged in 
 curves which end at the two poles. These curves mark what 
 Faraday called the lines of force of the magnet. Described 
 geometrically, a line of force is a .line in the field of force so 
 drawn that the tangent to it at any point lies in the direction 
 of the force at that point. The iron filings, by which the lines 
 of force around a magnet are marked out, are generally con- 
 siderably longer in one direction than in others. When they 
 fall on the paper they are magnetized by induction, the mag' 
 netic axis in each one generally coinciding with its greatest 
 length. The little magnets thus formed point along the lines 
 of force, and as they cling together by their mutual attraction, 
 they mark out some of the lines of force. 
 
 Faraday conceived -the lines of force around a magnet to 
 indicate a certain condition in the field, and to arise from cer- 
 tain conditions in the body of the magnet. He was therefore 
 led naturally to consider them as limited in number, and to 
 conceive that a certain definite number of them proceeded from 
 a pole of a certain strength, or that the strength of the pole 
 and the number of the lines of force which proceeded from it 
 were in proportion. Without adopting the physical hypothesis 
 of Faraday, we may use the conception of a limited number of 
 lines of force to describe the magnetic field with respect to the 
 force which is exerted at any point in it upon a unit pole. 
 This force at any point is called the strength of the field, or the 
 magnetic intensity at that point. Let us suppose that a certain 
 definite number of lines of force is associated with each mag- 
 netic pole of unit strength. The strength of any pole will then 
 be represented by the number of lines of force associated with 
 it. Now in drawing the diagram of lines of force, we proceed 
 as follows: We draw a surface in the field at right angles to 
 the lines of force, and we draw the lines of force through that 
 surface, so distributed that the number of lines which pass 
 through each unit of area of the surface is proportional to the 
 strength of field at the point where that unit area is situated. 
 The diagram is then completed by continuing these lines of
 
 268 MAGNETISM. 
 
 force to their ends. With this diagram the strength of field 
 anywhere in the field may be determined. For it may be shown 
 that the strength of field, at any point in the field, is then pro- 
 portional to the number of lines of force which will pass 
 through a unit area, set up at that point perpendicularly to 
 the lines of force. 
 
 When a piece of iron is placed in a matrnc- ic field, it be- 
 comes magnetized by induction. The intensity of its magnet- 
 ization depends on the strength of the field, and the direction 
 of the axis of the magnet formed by induction depends on the 
 direction of the lines of force of the fieid. We may state the 
 general rule for the direction of the axis, and the position of 
 the poles, by saying, that the positive direction of tne axis 
 coincides with the positive direction of the lines of force of the 
 field. By the positive direction of the axis, we mean the direc- 
 tion from the south to the north pole in the iron. This is the 
 direction which is taken as positive when we speak of the point- 
 ing of a compass needle. By the positive direction of the line 
 of force, we mean the direction in which a small magnet will 
 point if placed on that line. 
 
 A great deal of attention has been devoted to the study of 
 the magnetic field of the earth, partly for scientific reasons, 
 but mainly on account of its great importance in navigation 
 and surveying. The elements of the magnetic field which are 
 examined by experiment are the variation or declination, the 
 dip, and the horizontal intensity, that is, the horizontal com- 
 ponent of the strength of field. As to the method of determin- 
 ing the horizontal intensity, this much may be said. If a 
 small magnet is suspended so that it can swing freely in the 
 horizontal plane, and is then turned out of the magnetic merid- 
 ian, it will be acted on by a couple tending to bring it into the 
 magnetic meridian. For a swing of only a few degrees, the 
 moment of this couple is proportional to the angular devia- 
 tion from the magnetic meridian. The magnet therefore exe- 
 cutes oscillations which are similar to those of a pendulum, 
 and the time of oscillation is expressed by a formula like the 
 pendulum formula, in which, however, the moving force is not
 
 MAGNBTISM. 259 
 
 the weight of the body, but is due to the forces exerted by the 
 field upon the poles of the magnet. By determining the times 
 of oscillation of such a magnet at different places on the earth's 
 surface, the horizontal intensities at those places can be com- 
 pared. 
 
 The value of such determinations as these depends on the 
 constancy of the magnet with which they are made. By using 
 the same magnet to deflect another one from the magnetic 
 meridian, the strength of its poles and the horizontal intensity 
 are brought into another relation, and by a combination of 
 this relation with the one obtained from the former experi- 
 ments, a measure of the horizontal intensity can be obtained 
 in absolute units, and independent of the particular magnet 
 with which the experiments are made. 
 
 It is found that the horizontal intensity, like the variation 
 and the dip, undergoes both secular and daily changes. Not- 
 withstanding the great amount of study which has been devoted 
 to the earth's magnetism, no satisfactory theory of its origin 
 has yet been given. 
 
 162. Para magnetism and Diamagnetism. Hitherto we have 
 described and studied magnetism as if it were a property only 
 of iron. This, however, is not the case. Iron excels all other 
 bodies, in a very marked degree, in the extent to which it ex- 
 hibits magnetic properties, and it is apparently the only sub- 
 stance from which permanent magnets can be made. The other 
 metals of the iron group, and especially nickel, are also capable 
 of being magnetized by induction in the way in which iron is. 
 They are called paramagnetic bodies. A few other substances, 
 among them oxygen, either in the gaseous state or in the liquid 
 state, are paramagnetic. 
 
 Until 1846 it was supposed that all other substances were 
 not affected in the least in a magnetic field, but in that year 
 Faraday, by the use of a very powerful electromagnet, found 
 that all substances upon which he experimented were more or 
 less affected by the magnetic field, but in a different way from 
 that in which the paramagnetic bodies are affected. The effect 
 discovered by him is exhibited most strikingly by bismuth.
 
 260 MAOSKTISM, 
 
 The contrast between the behavior of bismuth and of iron is 
 shown by the following experiments: If a rod of iron is sus 
 pended between the poles of a strong magnet, it will turn and 
 point along the lines of force. The action is in accordance 
 with the general law of induction, which has already been 
 stated. On the other hand, if a rod of bismuth is suspended in 
 the same field, it will turn until it points across the lines of 
 force. Since the bismuth does not retain any traces of magnet- 
 ization when it is removed from the field, we are not able to 
 tell exactly what condition it assumes when in the field, but 
 its behavior in the field can be explained by supposing that it 
 becomes a magnet, so developed that the positive direction of 
 its axis is opposite to the positive direction of the lines of force 
 of the field. Bodies which exhibit this property are called 
 diamagnetic bodies. The action, even in the most pronounced 
 cases, is extremely feeble. 
 
 The hypothesis made in the last paragraph, that the mag- 
 netic arrangement in a diamagnetic body is opposite to that 
 which is set up in a paramagnetic body, is extremely difficult 
 to reconcile with our theories of magnetism. An experiment 
 tried by Faraday indicates a way by which that difficulty may 
 be avoided. Faraday filled a small glass tube with a solution 
 of sulphate of iron, which is a paramagnetic body. When this 
 tube was suspended in the magnetic field, it pointed along the 
 lines of force. When, however, it was immersed in a stronger 
 solution of sulphate of iron, it pointed across the lines of force. 
 This experiment indicates the following hypothesis to account 
 for the contrast between paramagnetic ana diamagnetic bodies ; 
 that the ether is itself a magnetizable body, and that para- 
 magnetic bodies acquire a greater intensity of magnetization 
 than the ether around them, while diamagnetic bodies acquire 
 a less intensity of magnetization. There is no way by which 
 these hypotheses can be tested by experiment. 
 
 163. Causes Affecting Magnetisation. We have used the 
 phrase, intensity of magnetization, to express the magnetic 
 condition of a body. This phrase must now be defined. In 
 order to do so, we first define the magnetic moment 01 a mag-
 
 MAGNETISM. 261 
 
 net, whether permanent or induced, as the product of the 
 strength of one of its poles and the distance between its poles. 
 Its intensity of magnetization is then defined as the quotient 
 of its magnetic moment, divided by its volume, or the magnetic 
 moment of unit of volume. It will easily be seen that, when 
 the unit of volume is taken small enough, this quantity char- 
 acterizes the magnetic condition of each part of the magnet. 
 The definition assumes that the magnetism exhibited by the 
 magnet is due to a condition which exists in all parts of it. 
 This assumption is suggested by the experiment of Gilbert 
 already referred to, and is justified by the facts which are now 
 to be cited. 
 
 When a piece of soft iron is brought into a magnetic field, 
 it is immediately magnetized by induction. Its intensity of 
 magnetization does not reach its full value instantly. If the 
 strength of the field is gradually increased, the intensity of 
 magnetization also increases. The two quantities at first in 
 crease proportionally, but as the strength of the field increases, 
 the intensity of magnetization for a time increases faster and 
 then more slowly than the strength of the field, and at last 
 reaches, or nearly reaches, a limit. The iron is then said to be 
 saturated. If the strength of field is now diminished until it 
 gradually becomes zero, the intensity of magnetization dimin- 
 ishes, but does not entirely vanish. In order to reduce it to 
 zero, the field must be reversed, and its strength in the reverse 
 direction raised to a certain value. After it has passed this 
 value, the intensity of magnetization also reverses, and in- 
 creases in the reverse direction until the iron is again satu- 
 rated. By a repetition of this process, the magnetization of 
 the iron can be carried through a cycle of values. For each 
 value of the strength of field there will be two values of the 
 intensity of magnetization, depending on whether the particu- 
 lar strength of field has been reached by increasing, or by 
 decreasing, the field strength. This general behavior of iron, 
 by which its magnetization seems to lag behind the magnet- 
 izing force, is called by Ewing, who investigated it, hysteresis.
 
 262 MACINKTISM. 
 
 If a piece of steel is placed in a magnetic field, it will also 
 become a magnet, though not so rapidly, nor to such an extent, 
 as an equal piece of iron. Its magnetization can be enhanced 
 by striking it so as to make it ring. When it is removed from 
 the magnetic field, it does not lose its magnetism, when struck 
 or jarred, as the iron does, but retains at least the greater part 
 of it. It is thus a permanent magnet. By striking it again, 
 its magnetization will be diminished. If it is heated, its mag- 
 netization will also diminish, but will return to its original 
 value, or nearly so, when the magnet is cooled, unless the heat- 
 ing has been carried beyond a certain limit. If the tempera- 
 ture of the magnet is carried above 785 C., the magnetization 
 entirely disappears, and is not restored on cooling. A piece of 
 iron raised above that temperature and brought into a mag- 
 netic field exhibits no magnetic properties whatever, until its 
 temperature falls below that temperature. 
 
 164. Theories of Magnetism. When the actions of magnets 
 were first studied, they were ascribed to the presence in the 
 magnet of certain effluvia, or magnetic fluids, which were 
 assumed to be of two opposite kinds, possessing the properties 
 of repelling fluid of the same kind and of attracting fluid of 
 the opposite kind. These fluids were supposed to be present 
 in equal quantities in any mass of iron, and to be separated by 
 the process of induction. The fluid collected in one end of the 
 magnet formed the north pole, and that in the other end, the 
 south pole. Gilbert's discovery that when a magnet wag 
 broken, the two parts did not contain simply a north and a 
 south pole respectively, but became complete magnets, made it 
 necessary to suppose that these fluids existed in each particle 
 of iron, and were separated in it by induction, so that each 
 particle in the magnetized iron became a magnet. The theory 
 of magnetism was developed in this form by Poisson, and 
 showed itself capable of accounting for the laws oi magnetic 
 force, the phenomena of magnetic induction, and of the dis- 
 tribution of magnetic force in a magnet. 
 
 This theory does not explain so readily the facts which have 
 been described in the last section. In order to explain them,
 
 MAGNETISM. 263 
 
 Weber proposed another theory, which, as modified by Ewing, 
 is the one now generally accepted. It does not attempt to 
 account for magnetism itself, but only to explain the behavior 
 of magnetized bodies. It is rather a theory of the structure of 
 a magnet than a theory of magnetism. Weber supposed each 
 molecule of iron to be a magnet. In the unmagnetized condi- 
 tion of the iron these molecular magnets have no definite ar- 
 rangement. The magnetization of the iron consists in arrang- 
 ing the molecular magnets in chains or rows. If we consider 
 the magnetic forces which a single row of molecular magnets 
 thus arranged will exhibit, it appears that the contiguous poles 
 of two neighboring molecules will mutually destroy each other's 
 effects, so that the only poles which will be effective in produc- 
 ing an external magnetic field will be the north pole of a 
 molecule at one end of the row and the south pole of a molecule 
 at the other end. If we consider a magnet to be a collection of 
 such rows laid side by side, and take into account the fact that 
 the mutual repulsion of the similar free poles at the ends of 
 the rows will tend to make those ends diverge from each other, 
 we can account for the existence of magnetic poles, for the dis- 
 tribution of magnetic force over the ends of a magnetized bar, 
 and for the fact that the intensity of magnetization of a mag- 
 net is greater in the middle than it is at the ends. The effect 
 produced by striking a magnet is also explained by this theory ; 
 for, we may suppose that any one molecule of the iron is 
 restrained somewhat from pointing in the direction of the lines 
 of force of the field by the mechanical obstruction of the mole- 
 cules around it, and that if it is momentarily freed from this 
 obstruction by the jar produced by striking the magnet, it will 
 swing into position more readily ; or if it is already in position, 
 and the magnet is not in a magnetic field, will swing more or 
 less out of position, so as partly to disintegrate the rows. The 
 effect produced by heating the magnet, which gives its mole- 
 cules more freedom of motion, is explained in a similar manner 
 To explain hysteresis we must consider, with Ewing, that 
 the arrangement of an assemblage of molecular magnets is not 
 an entirely unordered or irregular one, but that by their
 
 264 MAGNETISM. 
 
 mutual attractions the molecules will arrange themselves in 
 stable groups. When a magnetizing force is applied to a col- 
 lection of such groups, the first effect will be to turn the mole 
 cules of the group toward the line of the force, and so to 
 increase the intensity of magnetization. When the magnetizing 
 force reaches a certain limit, the groups, one by one, cease to 
 be stable. The molecules in them then turn freely under the 
 magnetizing force, and form new stable groups, of which the 
 intensity of magnetization is much greater. This description 
 is in accord with the experimental fact that the magnetization 
 of iron undergoes a great increase for a comparatively small 
 change of the magnetizing force, between certain limiting 
 values. After these new groups are formed, the further 
 increase of the magnetizing force merely brings the molecules 
 of the groups more nearly parallel to each other. Complete 
 saturation will be reached when they are all arranged in 
 parallel rows. When the magnetizing force is again dimin- 
 ished, the stability of the groups that have been formed con- 
 tinues even after the value of the magnetizing force is less than 
 that at which the groups were formed, and the groups do not 
 break down and assume their original condition until the 
 magnetizing force is considerably diminished. 
 
 It is evident that this theory of Ewing explains all the 
 facts which we have explained by the theory of Weber. 
 
 The subject of magnetism is intimately connected with that 
 of the electric current, and will be considered again when we 
 deal with the properties of the current.
 
 STATIC BLECTBICITT. 265 
 
 STATIC ELECTRICITY. 
 
 165. Electric Attraction. If a piece of amber is gently 
 rubbed on a dry cloth, and is then brought near small bits of 
 straw or paper, these light bodies will be attracted by it, and 
 will adhere to it. This observation is a very ancient one. It 
 is said to have been known to Thales (600 B. C.). By the 
 ancient as well as by the mediaeval philosophers, this action 
 was looked on as something occult, and was supposed to be 
 peculiar to amber, just as the attraction of iron was peculiar 
 to the lodestone. It was not investigated in a scientific way 
 until it was studied by Gilbert, in connection with his study 
 of the magnet. Gilbert mounted a light pointer on a needle, 
 so that it could turn about freely, and with it as an indicatoi 
 he examined other bodies besides amber, to see whether they 
 would exhibit a similar attractive power. Sulphur had appar- 
 ently been found by some previous observer to behave like 
 amber, and Gilbert found that many other bodies behaved in 
 a similar way. From the Greek name for amber, electron, he 
 called this action electric action. We now use the word elec- 
 tricity to express the assumed cause of this action. A body 
 exhibiting electric action is said to be electrified, or to be 
 charged with electricity. 
 
 Von Guericke made an important observation by noticing 
 that the light bodies, which were attracted to the electrified 
 body, were repelled by it after a time. 
 
 166. Electric Conductors. In experimenting on electric 
 action, Grey discovered that when certain bodies were brought 
 in contact with an electrified body, they also acquired the 
 power of attracting light bodies. He next undertook to trans 
 mit this power from the electrified body to another body 
 through a long thread. In his first experiment, he hung an 
 ivory ball by a thread from an upper window, and found that 
 when the electrified body was touched to the upper end of the 
 thread, the ball became electrified. He tried next a similar
 
 260 STATIC ELECTRICITY. 
 
 experiment, in which the thread was suspended horizontally 
 by loops of thread, and found that in this case the ball was not 
 electrified. Considering the reason of his success in the one 
 case and his failure in the other, he conceived that it might be 
 due to the conducting away of the electricity by the loops on 
 which the thread was suspended in the second experiment. He 
 accordingly suspended the thread by other bodies, and found 
 at last that, when silk thread for the suspension was used, the 
 action was transmitted to the ball. Grey thus proved the dis- 
 tinction between conductors and non-conductors. Many bodies, 
 of which the metals are the most conspicuous, transmit the 
 electric condition from one body to another, and are classed as 
 conductors. Other bodies, of which glass, the resins, and silk 
 are examples, transmit this condition either very slowly, or 
 not to any perceptible degree, and are classed as non- 
 conductors. Subsequent investigation has shown that there is 
 no sharp line of separation between these two classes of bodies. 
 Probably all bodies conduct to some extent. The distinction,, 
 however, between conductors and non-conductors, is generally 
 made, and many substances are so nearly non-conductors, that 
 they can be treated as such in most experiments. 
 
 107. Electric Attractions and Repulsions. Putting to- 
 gether the transmission of the electric condition, as observed 
 by Grey, and the repulsion of light bodies from the electrified 
 amber after contact with it, as observed by Von Guericke, it 
 was easy to see that the repulsion might be due to the similar 
 electric condition of the bodies. The experiment was there- 
 fore suggested of examining the way in which various electri- 
 fied bodies act on each other. Such experiments were carried 
 out by Dufay in 1733, and a few years later by Franklin. 
 These investigators found that all bodies which can be electri- 
 fied by friction can be placed in one of two classes, according 
 as they repel or 'attract an electrified piece of glass. These 
 experiments are most easily conducted by using as an indi- 
 cator a gilded pith ball hung by a non-conducting silk thread. 
 If a rod of glass is electrified by friction, and brought near 
 this ball,, it will at first attract the ball to it. As soon, how-
 
 STATIC KLECTRICITY. 267 
 
 6ver, as the ball acquires the electric condition by contact with 
 the glass, it is repelled from the glass. Its electric condition 
 is then the same as that of the glass, and it can be used in- 
 stead of a piece of glass in testing other bodies. According to 
 Dufay, it is vitreously electrified, or charged with vitreous 
 electricity. According to Franklin, who adopted another the- 
 ory and considered the effects to be described as due to the 
 excess or defect in the body of a single substance, it is posi- 
 tively electrified. Franklin's terms are those now generally 
 used, although the views which led to them are no longer held. 
 
 To test the electric relations of various bodies, they are 
 submitted to friction, and then presented to the positively 
 electrified indicator. Some of the bodies thus treated will 
 repel the ball as the glass did, and are therefore in the same 
 electric condition as that of the glass. The other bodies will 
 attract the ball. The question arises whether this attraction 
 is similar to that exerted by the glass on the ball in the first 
 instance, or whether it is something different. To test this, 
 the ball is touched with the hand, so that its electric condition 
 is removed. One of the bodies which causes attraction of the 
 positively electrified ball is then presented to it. The ball 
 behaves exactly as it did when the glass was brought near it; 
 that is, it is first attracted, until it comes in contact with the 
 body, and is then repelled. It is therefore presumably in the 
 same electric condition as that of the body which it has 
 touched. In this condition it is repelled by all the bodies 
 which attracted it when it was positively electrified, and is 
 attracted by all the bodies which then repelled it. The ball 
 has therefore acquired another condition, in which the forces 
 which it exerts are opposite to those which it exerts when 
 positively electrified. This other condition is developed on 
 resin, and Dufay therefore said that the ball in this condition 
 is resinously electrified, or is charged with resinous electricity. 
 In accordance with his theory. Franklin said that it is nega- 
 tively electrified. This is the term which we now use to ex- 
 press this condition. 
 
 Dufay summed up the results of his investigation in the 
 law that there exist two electric conditions, such that two
 
 ZOO STATIC ELECTRICITY. 
 
 bodies in the same condition repel each other, while two bodies 
 in different conditions attract each other. This law may be 
 otherwise stated by saying that similarly electrified bodies 
 repel each other, while dissimilarly electrified bodies attract 
 each other. As we shall subseqeuntly find that the super- 
 position of dissimilar electricities on the same body results in 
 the disappearance of both of them, we may also speak of oppo- 
 sitely, instead of dissimilarly, electrified bodies. 
 
 168. The Electric Spark. Von Guericke constructed an 
 electric machine by mounting a ball of sulphur so that it could 
 be turned around an axis, and setting near it a metallic body 
 supported by a non-conductor. When the ball was turned, and 
 the dry hands, or a piece of flannel, were pressed upon it, it 
 became negatively electrified, and imparted negative electricity 
 to the metallic conductor. With this machine, Von Guericke 
 found that, after the conductor had been receiving electricity 
 for some time, a spark would pass from it to any conducting 
 body held near it. 
 
 Other electric machines were soon constructed which were 
 far more efficient than this primitive one, and with them much 
 larger sparks were obtained. These sparks were generally 
 taken to be evidence of the passing of something, called elec- 
 tricity, from the charged body. 
 
 169. The Leyden Jar. Cunaeus of Leyden attempted to 
 collect electricity in water by holding a glass of water near the 
 conductor of an electric machine, and allowing sparks to pass 
 through a nail partly immersed in the water. After the sparks 
 had passed for some time he attempted to withdraw the nail, 
 and in so doing received a shock through his arms and body. 
 In the discovery of this effect he had been anticipated by von 
 Kleist, but Cunaeus' account of his discovery attracted atten- 
 tion, and the instrument by which the shock could be given 
 was called the Leyden jar. This name is still in use, although 
 the Leyden jar is simply one example of the class of instru- 
 ments known as electric condensers. 
 
 In its present form the Leyden jar is a glass jar coated 
 within and without, over its bottom and the lower part of its
 
 STATIC ELECTRICITY;. 269 
 
 sides, with tinfoil. Through the stopper in the neck is fixed a 
 metallic rod, ending above in a ball or knob, and connected 
 with the inner coating of the jar by a chain. To charge the 
 jar the outer coating is touched with the hand, or is otherwise 
 brought in conducting contact with the earth, and sparks are 
 allowed to pass to the knob. After they have passed for some 
 time, the jar may be discharged by touching one end of a con- 
 ducting body to the outer coating, and bringing tue other end 
 of it near the knob. A spark will pass between the knob and 
 the 'conductor, which is brighter and thicker and makes a 
 louder report than any of the sparks by which the jar was 
 charged. The successive charges which pass to the jar from 
 the machine seem to have been -accumulated or condensed in 
 the jar. When the discharge is taken through the body, the 
 shock is often very severe. 
 
 The explanation of the action of the Leyden jar was first 
 given by Franklin. In order to appreciate it, we must con- 
 sider the way in which an electrified body acts on bodies near 
 it, to produce the electric condition in them. 
 
 170. Electric Induction. In examining the effects of an 
 electrified body on other bodies, it is necessary to mount these 
 bodies on non-conductors, or to insulate them. When a con- 
 ductor thus insulated is brought near an electrified body, it 
 will exhibit signs of electrification. We test the peculiarities 
 of its electrification by the use of a proof-plane. This is a 
 small disk of sheet metal mounted on a non-conducting handle. 
 When the proof-plane is laid on the surface of the conductor 
 at any point, it acquires a charge similar to that of the con- 
 ductor at that point, and by removing it and testing the 
 charge on it, the charge on the conductor can be determined. 
 By the use of such a proof-plane it is shown that the parts of 
 the conductor already described which are farthest from the 
 electrified body have a charge like that of the electrified body, 
 and that those parts which are nearest the electrified body 
 have a charge of the opposite kind. These charges are shown 
 to be equal in amount by removing the conductor from the 
 presence of the electrified body, for then no charge can be de-
 
 270 STATIC ELECTRICITY. 
 
 tected on it. The action here described, by means of which an 
 electrified body can excite the electric condition in a neighbor- 
 ing body, which is insulated from it, is called electric induc- 
 tion. 
 
 The attraction between a charged body and any other body 
 brought near it is due to the attraction between the original 
 charge of the first body and the neighboring charge of the 
 opposite kind induced in the second body. When the second 
 body is insulated and contact occurs between the bodies, this 
 charge of opposite kind is destroyed by union with some of the 
 original charge and both bodies become charged similarly. In 
 this condition they repel each other. 
 
 If a conductor, when in the presence of a charged body and 
 charged by induction with both kinds of electricity, is touched 
 by the hand, or is connected with the earth by any conductor, 
 the charge which it has on the end farthest from the electrified 
 body, and which is of the same sort as that of the electrified 
 body, disappears. The other charge, on the end nearest the 
 electrified body, remains unchanged. These two charges, there- 
 fore, seem to differ, in that one of them is free to leave the 
 conductor by passing through any conductor presented to it, 
 while the other is bound or retained in the conductor. They 
 are called the free and the bound charge respectively. The 
 experiments here detailed show that the bound and the free 
 charges are equal in magnitude, and that the bound charge is 
 the one which is opposite in kind to the original charge. 
 
 To apply these facts to the explanation of the behavior of 
 the Leyden jar, we notice that, when the jar is being charged, 
 the inner coating receives a charge directly from the machine, 
 and acts as the original electrified body. By the action of the 
 charge which it first receives, the outer coating is charged by 
 induction, and its induced free charge passes off to the earth. 
 Its bound charge acts by induction on the inner coating, so as 
 to make a free charge on that coating of the opposite sort to 
 that which is supplied by the machine. The machine therefore 
 can supply an additional charge to neutralize this, and the 
 charge on the inner coating is increased by the amount of the
 
 HTAT1C KLECTR1CITT. 271 
 
 bound charge which was developed in it. This increased 
 charge on the inner coating again acts by induction, to increase 
 the bound charge on the outer coating, and this process goes on 
 until the jar discharges itself or until a limit is reached, 
 which is determined by the nature of the electric machine that 
 is supplying the charge. In the final condition of the jar when 
 charged, the conductor of the machine and the knob of the jar 
 are in similar electric conditions, so that no farther charge 
 can pass between them, and two opposite charges, of practi- 
 cally equal magnitude, confront each other on the surfaces of 
 the two coatings which are next, to the glass. When conduct- 
 ing connection is made between the outer and the inner coat- 
 ing, these charges recombine through the conductor, and the 
 jar is discharged. 
 
 The laws of electric induction are also illustrated in an 
 instrument invented by Volta, known as the electrophorus. It 
 consists of a sheet of sulphur, rubber, or some similar sub- 
 stance, which can be electrified by friction, called the plate, 
 and a metallic disk furnished with an insulating handle, 
 called the carrier. The plate is electrified by friction, generally 
 negatively, and the carrier placed upon it. In this position 
 the carrier is usually supported on slight prominences in the 
 plate, so that it is in contact with the plate at very few points, 
 and does not receive a charge from it by conduction. It is 
 charged by induction, the charge on its lower surface being 
 .positive, opposite to that of the plate. The negative charge on 
 its upper surface is free, and is removed by touching the car- 
 rier with the hand. When the carrier is lifted from the plate, 
 the bound positive charge, being no longer constrained by the 
 presence of the negative charge of the plate, distributes itself 
 over the carrier. This charge may then be communicated to 
 any conductor which it is desired to charge, and the carrier 
 may be charged again in a similar manner, as often as we 
 please, the original charge on the plate being retained by it 
 and remaining unaltered by the operation. 
 
 171. Distribution of Electricity. On* the hypothesis that 
 the electric charge is some sort of fluid whose parts repel each
 
 272 STATIC ELECTRICITY. 
 
 other, it ,is easy to see that the distribution of electricity in an 
 insulated conductor will depend upon the shape of the con- 
 ductor It was shown first by Franklin, and long afterwards 
 more conclusively by Faraday, that whatever be the shape of 
 the conductor, the charge in it resides entirely on its surface. 
 By testing different bodies with the proof-plane, it was also 
 shown that the density of the charge, that is, the. amount on 
 unit of surface, is greatest at the parts of greatest curvature. 
 When the law of electric force was discovered, the distribution 
 on bodies of particular forms was calculated. The experi- 
 mental examination of the distribution on such bodies verified 
 the results of these calculations. 
 
 The density of the charge is especially great at any part of 
 the conductor which ends in a sharp point. At such points, 
 the pressure of the dense charge against the surrounding insu- 
 lator is especially great,, and the charge therefore passes from 
 them with great facility. This property of points, of dis- 
 charging bodies and also of receiving charges, is made use of 
 in all sorts of electric apparatus. 
 
 . 172. Early Theories cf Electricity. After Dufay had dis- 
 covered the opposite effects produced by the charges on glass 
 and on resin, he developed a theory of electricity to explain 
 the facts. He supposed that there exist in the universe two 
 fluids, which he called vitreous and resinous electricity; that 
 in any uncharged body these fluids exist in equal amount ; that 
 they are separated from each other by friction, one of the two 
 bodies which are rubbed together retaining a surplus of the 
 one fluid, while the other body has a surplus of the other fluid ; 
 that the parts of either of these fluids repel other parts of the 
 same fluid and attract parts of the other fluid. This theory, 
 which is commonly known as the two-fluid theory of electricity, 
 was not at first accepted, but it gradually won its way into 
 prominence, and is the one which has determined most of the 
 nomenclature of electricity. Franklin adopted the view that 
 there exists in the universe a single fluid, called electricity, and 
 that a body when charged positively possesses a surplus of this 
 fluid, while a body charged negatively possesses less than the '
 
 STATIC ELECTRICITY. 273 
 
 normal amount. This theory was developed by Aepinus, and 
 for some time was generally held. It fell into disuse after a 
 while, in consequence chiefly of the necessity which Aepinus 
 found, in order to explain the electric attractions and repul- 
 sions, as well as the passivity of unelectrified bodies, of sup- 
 posing that matter repelled matter according to the same law 
 as that by which electricity repelled electricity. Without the 
 assumption of a mutual repulsion between the particles of 
 matter, as well as of an attraction between matter and elec- 
 tricity, the actual forces observed could not be explained. 
 
 The modern theories of electricity unite in assuming that 
 electricity, whatever it may be, is not a continuous fluid, but 
 exists in separate portions, or units, which may be called elec- 
 tric atoms or electrons. 
 
 173. Law of Electric Force. For any further development 
 of the subject, it becomes important to know the law of the 
 force exerted by one electric charge on another. The law was 
 first investigated by Cavendish about the year 1773. His 
 method of investigation depended on a theorem first proved by 
 Newton, to the effect that the strength of field at any point 
 within a uniform spherical shell, composed of any substance 
 exerting force which varies with the distance according to the 
 law of inverse squares, will be zero. By the most careful ex- 
 periments which he could make with the crude apparatus at 
 his disposal, Cavendish could not detect any evidence of any 
 electric force inside a sphere charged with electricity. He 
 therefore concluded that the elements of electricity act on each 
 other with forces which vary inversely with the square of the 
 distance. The determination of this law, by this method, in- 
 volves the assumption that the force exerted between two quan- 
 tities of electricity is proportional to their product. Cavendish 
 never published his proof of this law, and it remained unknown 
 until his papers were edited by Maxwell. The general method 
 used by Cavendish has been employed with the most sensitive 
 apparatus, and the law of electric force has been determined 
 with great exactness. 
 
 The law of electric force became known from the work of 
 Coulomb in 1785. In his investigations, Coulomb employed the
 
 274 STATIC ELECTRICITY. 
 
 torsion balance already described in 159. By means of this 
 instrument, he measured the repulsions between two similarly 
 charged spheres at different distances from each other, and 
 found that the force between them varied inversely with the 
 square of the distance between their centres. It had been 
 proved by Newton, that the force at any point outside a spher- 
 ical shell made of a substance which exerts force according to 
 the law of inverse squares, will be inversely as the square of 
 the distance of the outside point from the centre of the sphere. 
 The measurements of Coulomb were therefore consistent with 
 the hypothesis that each element of the electric charge repels 
 every other element with a force which varies inversely as the 
 square of the distance. A similar law was proved for the 
 .Attraction between two spheres oppositely charged. 
 
 By observing the force between two electrified spheres at a 
 definite distance, and comparing it with the force exerted at 
 the same distance between the spheres when the charge on one 
 of them had been halved by touching the sphere to another 
 of the same size, Coulomb showed that the force is propor- 
 tional to the magnitudes of the charges. 
 
 The law of electric force may therefore be stated by saying 
 that the force between two small electric charges is propor- 
 tional to the product of the charges and inversely proportional 
 to the square of the distance between them. Owing to the fact 
 that the force between two charges depends on the medium 
 which surrounds them, we assume in this definition that the 
 charges are in a vacuum. The force between the charges in 
 air is only slightly smaller than that in vacuum. 
 
 174. Quantity of Electricity. In measuring quantities of 
 electricity we find ourselves in the same position as when we 
 measure quantities of magnetism. We do not perceive elec- 
 tricity directly, or by any other effects than the forces which 
 it exerts. We are therefore compelled to measure it by means 
 of those forces. If we represent two electric charges by e and 
 e', the distance between them by r, and the factor of proportion 
 by fe, the equation which expresses the force between these 
 
 charges is F=fc . We can measure the force and the dis-
 
 STATIC ELECTRICITY. 275 
 
 tance, but until some convention is made, we cannot measure 
 the charges. It is conceivable that a charge might be arbitrar- 
 ily selected as a standard, with which other charges might be 
 measured by the examination of the forces which they exert on 
 the standard charge. But the impossibility of preserving such 
 a charge unaltered, and of using it in the study of other 
 charges without error, makes it necessary for us to determine 
 the unit charge in some other way. If we were to use this 
 method, the factor k would have a definite value, which might 
 be determined by experiment as the constant of electric force. 
 But so long as the unit charge is left undetermined, the value 
 of k is not fixed. By assuming an arbitrary value for it, we 
 fix the product of the two charges which act on each other. To 
 make this value as simple as possible, we set fc=l. When this 
 choice is made, the value of the product ce' is determined from 
 the known values of the force and the distance. If the charges 
 are equal, the value of either charge is determined from the 
 same data. In particular, we may define the unit electric 
 charge as that charge which will repel an equal and similar 
 charge at unit distance with unit force. In the C. G. S. system 
 the unit electric charge is one which will repel an equal and 
 similar charge at the distance of one centimetre with the force 
 of one dyne. 
 
 175. Electric Fields. In order to study the electric forces 
 exerted by any system of electrified bodies, it is convenient to 
 suppose that we can make use of a unit charge, which is per- 
 fectly insulated, and the force upon which we can always meas- 
 ure. This charge may be called a test unit. When such a test 
 unit is brought near an electrified body, it is generally acted 
 on by an electric force. The region in which such a force can 
 be detected is called an electric field. 
 
 If the directions of the forces at different points in an elec- 
 tric field are examined by means of the test unit, it will be 
 found that they vary from point to point in such a way that 
 they may be represented by drawing lines of force through the 
 electric field. A line of force is a line such that the tangent to 
 it at any point indicates the direction of the electric forae at
 
 276 STATIC ELECTRICITY. 
 
 that point. The positive direction of a line of force is the 
 direction in which the test unit will tend to move. It will be 
 shown later that the lines of force in any electric field begin at 
 positive charges and end at negative charges. We shall also 
 find it possible to connect the number of lines of force with the 
 charges at which they arise in such a way that the distribution 
 of the lines in the field indicates the force on the test unit at 
 each point. This force at any point is called the strength of 
 the electric field at that point, or the electric intensity, or 
 sometimes the electric force. 
 
 When an insulated conductor is placed in an electric field, 
 it becomes electrified by induction. We may describe the dis- 
 tribution of the charge in it, in a general way, by saying that 
 both the positive and the negative charges move in the con- 
 ductor under the action of the electric field as if they were 
 free. The positive charge is therefore displaced in the positive 
 direction of the lines of force, the negative charge, in the nega- 
 tive direction. This displacement continues until the forces 
 set up by the displaced charges, combined with the electric 
 force of the field, neutralize each other at every point within 
 the conductor. When this condition is reached, the displace- 
 ment of the charges ceases and the system is in electric equi- 
 librium. 
 
 176. Electric Potential. In his study of the law of gravi- 
 tation, Laplace introduced a certain function, by which his 
 analysis was very greatly facilitated. A similar function may 
 be used in the study of any forces which vary according to the 
 law of inverse squares. Such a function was applied to the 
 study of electricity by Green and was called by him the poten- 
 tial function. In the case of electricity the potential function 
 is found to be especially useful, and to be applicable even in an 
 elementary treatment of the subject. Indeed our modern study 
 of electricity depends upon its use. It is accordingly necessary 
 at this point to examine its most important properties. We 
 shall use the word potential to designate not only this func- 
 tion, but the property of the electric field characterized by it. 
 
 If a test unit is placed at a point in an electric field, and
 
 STATIC BLKCTBICITY. 277 
 
 is moved from that point to an infinite distance, work is done 
 upon it by the forces of the field during its motion. Analysis 
 proves that the work which is thus done depends only on the 
 position of the point at which the motion begins, and is inde- 
 pendent of the path through which the test unit moves. The 
 work thus done is equal to and measures the potential at the 
 point. If the work done by the forces of the field during the 
 motion is positive, the potential at the point is positive; if the 
 work done is negative, the potential is negative. 
 
 If two points in the electric field are chosen, the work 
 done on the test unit, as it moves from one of them to the 
 other, is the difference of potential between those points. 
 This difference of potential is independent of the path tra- 
 versed by the test unit during the motion. This may be shown 
 by analysis, and must otherwise be true if the principle of the 
 conservation of energy holds for the electric field. For if it 
 were not true, and if more work were done on the test unit 
 in one path than in another, an unlimited supply of work 
 might be obtained by allowing the test unit to move, under the 
 action of the field, over the path in which the greater amount 
 of work is done, and bringing it back, against the action of 
 the field, over the other path, and by repeating this cyclic pro- 
 cess as often as we please. This result is of course incon- 
 sistent with the principle of the conservation of energy. 
 
 If the distance between the two points, between which the 
 test unit moves, is very small, the force on the test unit will 
 be appreciably constant during the motion. If we represent 
 the force on the test unit in the direction of the motion by 
 F, the small distance traversed by s, and the potentials at the 
 two points by V and V, we obtain, from the relation between 
 the work done on the test unit and the difference of potential, the 
 equation Fs=V V. From this equation we obtain for the 
 
 V- V. 
 force in the direction of the motion the expression F= 
 
 The quantity on the right is evidently the change of potential 
 in the direction of the motion per unit of length, or is the rate 
 of change of potential in the direction of motion. In the 
 electric field the force is positive, or is in the direction of the
 
 "2~S STATIC KLKOTRICITT. 
 
 motion, when the unit moves from the point of higher poten- 
 tial to the point of lower potential. When there is no differ- 
 ence of potential between the neighboring points, there is no 
 force acting in the line joining them. In an electric field a 
 system of surfaces can be drawn, which are everywhere at 
 right angles to the lines of force. There is no component of 
 force lying in any of these surfaces, and consequently in no 
 one of them is there any difference of potential. Any such 
 surface, in which the potential has everywhere the same value, 
 is called an equipotential surface. From this relation it is 
 evident that any field of force, which can be mapped out by 
 lines of force, can also be mapped out by equipotential sur- 
 faces. 
 
 The peculiar applicability of the potential to the study of 
 static electricity is due to the fact that electricity moves 
 freely through a conductor so long as there is any electric 
 force acting on it, and attains equilibrium only when its dis- 
 tribution is such that no electric forces are acting within the 
 conductor. In this condition, as is evident from the relation 
 between force and change of potential, the potential of the 
 conductor must be everywhere the same, and the surface of 
 the conductor must be an equipotential surface. The potential 
 at any point on this surface is called the potential of the 
 conductor. 
 
 In order to measure the potential of a conductor accord- 
 ing to the definition of it which has been given, it would be 
 necessary to carry a test unit from that conductor to infinity. 
 By this operation we measure the difference of potential be- 
 tween the conductor and an infinitely distant point. Of 
 course this operation is impossible, and it is therefore neces- 
 sary, for practical purposes, to adopt the potential of some 
 other point as the standard potential, with which other 
 potentials shall be compared. The standard potential chosen 
 is the potential of the earth, which, it may be shown, re- 
 mains appreciably constant at all times, whatever be the elec- 
 tric operations which take place during our experiments. 
 This standard potential we take as the zero of potential.
 
 STATIC ELECTRICITY. 
 
 279 
 
 With this convention the potential of a conductor is the work 
 which is done on a test unit as it moves from the conductor to 
 the earth. The potential of a positively charged body is posi- 
 tive; that is, the work which is done on the test unit, aa it 
 moves from that body to the earth, is positive work done by 
 the electric forces. The potential of a negatively charged body 
 is negative; that is the work done by the electric forces on the 
 test unit, as it is moved from the body to the earth, is nega- 
 tive. These statements do not apply without exception to 
 bodies on which there are induced charges. 
 
 The tendency of a positive charge is to pass from a place 
 of higher potential to a place of lower potential. The ten- 
 dency of a negative charge is to pass from a place of lower 
 potential to a place of higher potential. When a caarged 
 body is brought near enough to a conductor whose potential 
 differs from its own, its charge will pass in the form of a 
 spark to the other conductor, until the potentials of the two 
 conductors become the same. 
 
 We may illustrate the application of the idea of potential 
 by describing, in terms of it, the use of- the electrophorus. 
 When the plate of the electrophorus is charged by friction, 
 its potential, and that of the region around it, becomes nega- 
 tive. The numerical value of this negative potential is high- 
 est at the plate, and diminishes from the plate to zero. When 
 the carrier is placed on the plate, it is brought into this 
 region of negative potential, and if it were not a conductor, 
 the potential on its lower surface would be negatively higher 
 than the potential on its upper surface. There would thus 
 exist in it a difference of potential, and the lines of force cor- 
 responding to this difference would be directed from its upper 
 surface to its lower surface. Such a force, however, cannot 
 exist in a conductor, and a positive charge is developed on the 
 lower surface, and a corresponding negative charge on the 
 upper surface, until the forces to which they give rise 
 neutralize the force which has been described, or until the 
 potential within the carrier is everywhere the same. When 
 the carrier is touched and so joined to earth, its potential is
 
 280 STATIC ELECTRICITY. 
 
 raised to zero, the potential of the earth. This involves the 
 loss of the negative charge of the carrier. The potential of 
 the carrier is then due to the superposition of the opposite 
 potentials due to the charge on the plate and the positive 
 charge on the carrier. When the carrier is lifted, work is done 
 on it against the attraction between these charges, and the 
 work thus done raises the electric energy of the carrier, and 
 gives it positive potential. 
 
 The potential of a conductor is measured by the work done 
 on unit charge, as it is carried from the conductor to the earth. 
 The potential of a conductor is therefore equal to unity, or the 
 conductor is at unit potential, when the work done on the 
 unit charge is the unit quantity of work. In the C. G. S. 
 system the potential of a conductor is unity when the work 
 done on the unit charge of that system, as it passes from the 
 conductor to the earth, is equal to one erg. 
 
 177. Equality of the Two Kinds of Electricity. It was 
 assumed by Dufay and Franklin, and accepted by all other 
 students of the subject, that the two electric conditions, called 
 vitreous and resinous, or positive and negative, existed in 
 every natural body in equal quantities. Our attention has 
 already been called to this assumption in our study of in- 
 duction, in Avhich we saw that the two charges which are 
 separated by induction will exactly neutralize each other when 
 the conductor is removed from the electric field. 
 
 It may also be shown by experiment that when electricity 
 is produced by friction, it appears in equal quantities of the 
 two kinds. For example, if a rod of glas is fitted with a cap 
 of silk or flannel, held in an insulating handle, and is turned 
 in the cap until its end is electrified, it may be shown that the 
 cap is also electrified oppositely; and the equality of the two 
 charges is shown by the fact that when the cap is on the glass 
 rod, no electrical effects can be detected. This experiment suc- 
 ceeds best if the glass and silk are rubbed together inside a 
 closed metallic vessel insulated from the earth. As we shall 
 see in the next paragraph, the exterior of this vessel ought to 
 appear electrified unless the two charges developed are ex-
 
 STATIC ELECTRICITY. 281 
 
 actly equal. That two opposite charges are developed, is 
 shown by removing either the glass or the silk from the vessel, 
 the exterior of which then becomes either negatively or posi- 
 tively electrified. But when the glass and silk are in the 
 vessel together, no external electrification can be detected. 
 Indeed, any electric machine, or the most intricate methods 
 for developing electricity, may be worked inside the vessel 
 without electrifying its exterior at all. 
 
 The final demonstration of the equality between the posi- 
 tive and negative charges, and of the invariable relation be- 
 tween them, was given by Faraday. The experiment by which 
 Faraday did this is commonly called the ice pail experiment, 
 because the vessels which he used in it were ice pails. In its 
 simplest form, the apparatus consists of a metallic vessel, set 
 on an insulating stand, and of an insulated conductor, which 
 can be introduced into the interior of the vessel. The cover 
 of the vessel is put on after this conductor is introduced. 
 Some sensitive apparatus is provided by which the electrifica- 
 tion of the exterior of the vessel can be examined. The vessel 
 is first discharged by joining it to the earth for a moment, 
 and the conductor, charged positively, is introduced within it. 
 The vessel is then found to be charged positively.' If it is 
 then joined to earth for a moment, its positive charge dis- 
 appears. If the charged conductor is now removed from the 
 interior, the vessel exhibits a negative charge. Judging from 
 what we have already learned about the distribution of in- 
 duced charges, we may conclude that the effect of introducing 
 the charged conductor within the vessel is to develop a nega- 
 tive charge on its inner surface and a positive charge on its 
 outer surface. These charges are equal in magnitude, as is 
 proved by their neutralizing each other, if the charged con- 
 ductor is removed before the positive charge of the vessel 
 has been removed by joining it to earth. Now to determine 
 the relative magnitudes of the original charge on the con- 
 ductor and the opposite charge which it produces by induction, 
 we introduce the charged conductor as before, and remove the 
 induced positive charge. We then touch the charged conductor
 
 282 STATIC ELKCTRLCITY. 
 
 to the interior of the vessel, or join it to the interior by a 
 conducting wire. No signs of electrification appear on the 
 exterior, and when the conductor is now removed, no charges 
 can be detected either on it or on the vessel. Faraday con- 
 cluded from this result that the charge on a body induces a 
 charge of the opposite sort, and of equal magnitude to itself, 
 on the conductor or conductors which completely surround it. 
 
 Before the significance of this experiment was appreciated, 
 it was usual to speak of charged bodies which are at some dis- 
 tance from other conductors as freely electrified bodies, and 
 to ignore the opposite charges which they produce in neigh- 
 boring bodies by induction. It was only when the charged 
 body is very near other conductors, as in the case of the 
 Leyden jar, that the presence of the induced charge was 
 thought of as having any special significance. Faraday's ex- 
 periment showed, however, that there is really no such thing 
 as a freely electrified body, but that the electrification of one 
 body involves the electrification by induction of the conductors 
 which surround it. This general truth suggests the hypothesis 
 that the two equal charges which confront each other across 
 the non-conducting medium which separates them, are the two 
 ends of a condition which exists in the medium, and that the 
 medium is the true seat of the electric action. We shall find 
 additional support for this hypothesis when we study the way 
 in which different media affect the charge which a body can 
 receive. 
 
 178. Capacity of Conductors. We define the capacity of 
 a conductor as the quantity of electricity which is required to 
 raise the potential of the conductor from zero to unity, when 
 th potential of all surrounding conductors is maintained at 
 zero. The capacity of a conductor depends, at least in part, 
 on its shape and size, and on the distance between it and sur- 
 rounding conductors. If we examine the explanation which 
 has been given in 170 of the charging of a Leyden jar, it will 
 be seen that the mutual inductive action between the two 
 coatings of the jar will be greater when they are nearer to- 
 gether, and that consequently a larger charge will be received
 
 STATIC KLKOTR1CITY. 288 
 
 by the inner coating from the same source, when the coatings 
 are near together, than when they are farther apart. This 
 general conclusion may be formally demonstrated in the case 
 of the spherical condenser. The spherical condenser is a con- 
 ducting sphere insulated from a concentric spherical conduct- 
 ing shell. The outer shell is joined to earth, and, through a 
 small hole in it, contact is made by means of a wire between 
 the interior sphere and the source of charge. The potential of 
 the interior sphere is everywhere the same, and we can cal- 
 culate it by calculating the potential at its centre. It may 
 be shown that the potential at a point at the distance r from 
 
 a chanje of magnitude e is equal to - Now all parts of 
 
 the charge on the interior sphere are at the same distance 
 from the centre of the sphere, and the potential at that cen- 
 tre due to the charge c on the sphere is equal to - , in which 
 
 r represents the radius of the sphere. The potential at the 
 same point, due to the equal and opposite charge on the inner 
 
 surface of the exterior shell, is e ., in which ' represents 
 
 the radius of the shell. The actual potential at the centre of 
 the sphere, and therefore of the sphere itself, is the sum of 
 
 / T f T \ 
 
 these potentials, so that we may write P=e| ) The ratio 
 
 of the charge to the potential, or -^, is the capacity of the in- 
 
 r f r 
 
 terior sphere, as we have already denned it. As the formula 
 shows, it is equal to the product of the two radii divided by 
 their difference, and therefore increases very rapidly as the 
 difference of the radii is diminished. 
 
 A system of conductors like the spheres just described, or 
 like the Leyden jar, of which the capacity is very great, is 
 called a condenser. The conductor which is charged in a 
 condenser differs from other conductors merely in being so 
 situated, with respect to the conductors which surround it, 
 that a relatively large charge is required to raise it to unit 
 potential. 
 
 It is obvious that the conclusion which has been proved to
 
 284 STATIC ELECTRICITY. 
 
 hold for the spherical condenser, that its capacity depends 
 upon the distance between the charged body and the con- 
 ductors which surround it, will hold for conductors of any 
 shape. 
 
 179. The Dielectric. An isolated experiment tried by 
 Franklin led him to believe that the non-conducting medium 
 between the two coatings of a Leyden jar plays an important, 
 and perhaps the principal part, in the charging of the jar. 
 We now try this experiment with what is known as a dissected 
 jar, that is, a glass vessel or cup which is set inside a metallic 
 cup, to serve as an outer coating, and receives another metallic 
 cup furnished with a conducting rod, to serve as the inner 
 coating. Such a jar may be charged and discharged in the 
 ordinary way. If the inner coating is removed while the jar 
 is charged, it will be found to have only a very small charge on 
 it. It appears, therefore, that the charge of the jar is not 
 carried about on the inner coating. If this coating is re- 
 placed, the full charge may be obtained from the jar. The ex- 
 periment is more striking if we remove both the coatings and 
 replace them by others of similar shape. The jar is still found 
 to be charged, and the conclusion is irresistible that the charge 
 has resided somewhere in or on the glass. It is natural to 
 make the hypothesis that the process of electrification con- 
 sists in setting up in the glass some peculiar condition, which 
 terminates at the two conducting coatings. 
 
 When a non-conducting medium is used to separate a 
 charged conductor from the other conductors which surround 
 it, and when we are studying the conditions which exist in 
 that medium, or the influence of the medium upon the charge, 
 we call the medium the dielectric. The first study of the 
 properties of dielectrics was made by Cavendish (1771-1781), 
 but his work was not published at that time, and we owe our 
 knowledge of those properties to Faraday. Faraday's in- 
 vestigation was carried out by the aid of two precisely simi- 
 lar spherical condensers. In one of these the space between 
 the two surfaces was always filled with air, in the other it 
 was first filled with air, and afterwards with the different
 
 
 STATIC ELECTRICITY. 285 
 
 dielectrics which he studied. The experiment consisted in 
 charging the inner spheres of both these condensers to the 
 same potential, by joining them both at the same time to the 
 same source, and in discharging them in turn through an 
 instrument by means of which the quantities discharged could 
 be compared. As was to be expected from the similarity of the 
 condensers, the quantity discharged was the same from each 
 when both contained air as the dielectric. When sulphur or 
 parafline was used in one of the condensers as the dielectric, 
 the quantity obtained from it was many times greater than 
 the quantity obtained from the other. This experiment proved 
 that the capacity of a condenser depends on the nature of the 
 dielectric which separates its conducting parts. Faraday used 
 the term specific inductive capacity to represent the effect of 
 a particular dielectric, as determined by the ratio of the 
 capacity of a condenser in which it is used to the capacity of a 
 similarly shaped condenser in which air is used. This ratio, 
 which is a characteristic constant for each particular di- 
 electric, is now generally called the dielectric constant. 
 
 This direct and very complete evidence of the participation 
 of the dielectric in the process of charging a body confirmed 
 the hypothesis which has already been referred to, that the 
 production of the charge involves setting up in the dielectric a 
 peculiar condition, which terminates at the charged con- 
 ductors. This hypothesis has been worked out analytically 
 by Maxwell, and has been shown to give a complete account of 
 the relations of charged bodies. 
 
 On this view of the process of electrification, the lines of 
 force which may be drawn through the dielectric are naturally 
 taken to indicate lines along which the action in the dielec- 
 tric takes place. We may map out the electric field, and in- 
 dicate its intensity at various points, by drawing one line 
 of force, or any assumed number of lines of force, from each 
 unit of positive charge, through the dielectric, to the cor- 
 responding unit of negative charge. The measurements of the 
 electric force from which these lines are drawn must be made 
 in a narrow crevasse cut in the dielectric, with its faces per-
 
 286 STATIC ELECTRICITY. 
 
 pendicular to the lines of force. The force thus determined 
 is called the electrostatic induction, and the lines are called 
 lines of electrostatic induction. It may be shown that where 
 these lines of induction are most closely crowded together, the 
 electric intensity is greatest, and that the number of lines of 
 induction, which at any point pass perpendicularly through a 
 unit of area, will be proportional to the intensity at that 
 point. 
 
 180. Maxwell's Descriptive Theory of Electrification. In 
 connection with the theory of medium action, it is interesting 
 to examine the particular form given it by Maxwell, rather 
 as a description than as expressing any final theory of the 
 real condition in a dielectric. Maxwell supposed the ether, 
 and therefore all bodies in the ether, to be filled with elec- 
 tricity. We may best think of this electricity as existing in 
 separate portions. Maxwell supposed that this electricity 
 moves freely, or with a resistance due only to friction, 
 through conductors, but that its displacement in a dielectric 
 is resisted by a force which increases with the extent of the 
 displacement, and which he likened to elasticity. The process 
 of charging a body then consists, according to this description, 
 in a displacement of the electricity along the lines of force 
 until it is checked by the electric elasticity. Considering the 
 displacement with respect to the dielectric, it is inward when 
 the dielectric is bounded by a positively charged conductor 
 and outward when it is bounded by a negatively charged con- 
 ductor. When connection is made by a conductor between 
 two oppositely charged bodies, a flow of electricity passes 
 through it and the strain in the dielectric is relieved. The 
 work which is done in charging a conductor is, in this descrip- 
 tion, stored up in the dielectric as work done in effecting the 
 electric displacement. To account for the various dielectric 
 constants of different media, we suppose that the same force 
 will produce different displacements in the different media. 
 
 181. Electric Machines. The first electric machine, con- 
 structed by Von Guericke, has already been described. It 
 was very soon improved upon by using a glass plate or
 
 STATIC ELECTRICITY. 287 
 
 cylinder, instead of the sulphur ball, and by setting against it 
 a pad of flannel or leather, by the friction of which the glass 
 is electrified. An insulated conductor, called the prime con- 
 ductor, furnished with a comb or row of points, pointing to- 
 ward the glass, is used to collect the charge developed as the 
 machine is turned. Machines of this sort are called frictional 
 machines. They are now very little used. 
 
 The germ of all "modern electric machines is found in the 
 electrophorus of Volta. It is plain that if the operation of 
 charging and discharging the carrier, which we have described 
 as carried on by the hand, is executed rapidly by some mechan- 
 ism, a rapid succession of charges can be obtained. It is not 
 necessary to go into the details of these induction machines. 
 
 182. Electroscopes and Electrometers. Any apparatus 
 which will indicate the presence of an electric charge, and 
 which will enable us to determine its character, is called an 
 electroscope. Any light body, such as a straw suspended by 
 a silk thread, may serve for that purpose. A very common 
 form of electroscope is that known as the gold leaf electro- 
 scope. It consists of two narrow strips of gold foil, which 
 hang down from the end of a conducting rod, supported in the 
 neck of a glass bottle or flask. When this system is charged, 
 the gold leaves, being charged similarly, mutually repel each 
 other, and diverge from each other. They are most commonly 
 charged positively by induction, using a negative primary 
 charge. While the gold leaves are diverging because of their 
 positive charges, if the upper end of the rod to which they are 
 attached is brought near a positively charged body, an addi- 
 tional positive charge will be given to them by induction, 
 and they will diverge more widely. If, on the other hand, the 
 body to which the rod is presented is charged negatively, the 
 gold leaves will receive a negative charge by induction, which 
 will neutralize some of their positive charge, and they will 
 diverge less widely. This apparatus, without any charge, is a 
 very sensitive indicator of the presence of a charge on any 
 body to which it is brought near. 
 
 An electrometer is an instrument by which the difference 
 of potential between two bodies can be measured, or compared
 
 288 STATIC ELECTRICITY. 
 
 with some other difference of potential. The attracted-disk 
 electrometer, first used by Snow Harris, and developed by Lord 
 Kelvin, furnishes a measure in absolute units of difference of 
 potential. It consists essentially of a large horizontal disk, 
 above which a smaller disk is set at a known distance. Thia 
 smaller disk is supported by a balance or a spring, by means 
 of which the force upon it may be measured. By measuring 
 the force exerted by the charges on the disks, when they are 
 brought to different potentials, the difference of potential be- 
 tween them may be measured in absolute units.
 
 THE BLKCTRIC CURRENT. 
 
 THE ELECTRIC CURRENT. 
 
 183. Galvani's Discovery. In 1791 the Italian physiologist 
 Galvani happened to notice that the legs of a recently killed 
 frog, lying near an electric machine, were thrown into con- 
 vulsions whenever a spark passed from the machine. He in- 
 vestigated this phenomenon on the hypothesis that the nerves 
 and muscles act like the coatings of a charged Leyden jar, and 
 in following up this hypothesis he found that the convulsive 
 movements occurred whenever the lumbar nerves and the 
 muscles of the leg were joined by a metallic connection. The 
 action, which was comparatively slight when this connection 
 was made by one metal, was made much greater by touching 
 the nerve with a strip of one kind of metal, the muscles with 
 a strip of another kind, and then bringing the two strips to- 
 gether. Galvani interpreted this result consistently with the 
 hypothesis already stated, but other observers were led to 
 consider the frog's legs simply as a very sensitive electroscope, 
 and to ascribe the action observed to the contact of the metals. 
 
 184. Volta's Series. In the years 1798-1802 the Italian 
 physicist Volta succeeded in demonstrating the production of 
 electrification by the contact of metals, and in applying his 
 discovery to the construction of an apparatus for the pro- 
 duction of the electric current. By the use of the condensing 
 electroscope, which he had invented a few years before, Volta 
 was able to prove that when pieces of two different metals, like 
 copper and zinc, are brought in contact, the electric condition 
 of the one becomes different from that of the other. As we 
 should now describe it, the potentials of the two metals be- 
 come different from each other when the metals are brought in 
 contact. This difference of potential is very slight, altogether 
 too slight to be detected by an observation with any but a 
 sensitive electroscope. 
 
 Volta demonstrated that the difference of potential occur- 
 ring on contact is a definite characteristic difference for each
 
 290 THB KLECTR1C CUBRENT. 
 
 pair of metals employed. He proved also that the electric 
 effect produced by arranging a number of different metals in 
 contact with one another in succession is to charge the two 
 metals at the end of the row, just as they would be charged if 
 they were immediately in contact. By measuring the different 
 potential differences existing between the different pairs of 
 metals, Volta found that the algebraic sum of the potential 
 differences arising at the successive contacts is equal to the 
 difference of potential arising from the contact of the two 
 terminal metals of the arrangement. If the two terminal 
 metals are brought in contact, so that a closed series of dif- 
 ferent metals is formed, the algebraic sum of the differences 
 of potential arising from the contacts will equal zero. That 
 this is the case appears from the fact that there is no evidence 
 of any continuous movement of electricity, arising from an 
 unbalanced potential difference, in such a metallic circuit. 
 Taking some one metal as a standard, and determining the 
 potential differences arising by its contact with other metals, 
 we obtain data from which the difference of potential arising 
 from the contact of any two of the metals may be determined. 
 Such a set of data is called a Volta's Series, or the Electro- 
 motive Series. Volta used the term, conductors of the first 
 class, to designate those substances which give rise to such 
 potential differences that their algebraic sum equals zero, 
 when the substances are arranged in a closed circuit. 
 
 Volta could not discover any difference of potential aris- 
 ing from the contact of a metal with a liquid. He therefore 
 called the liquids conductors of the second class. We now 
 know that slight potential differences do arise from such a 
 contact, but that they are neither of such a magnitude nor of 
 such a sign as to reduce the sum of the potential differences 
 in a circuit to zero, when a conductor of the second class forms 
 a part of the circuit. In such a circuit, therefore, in which, 
 as its essential characteristic, two different conductors of 
 the first class are in contact with a conductor of the second 
 class, there exists an unbalanced difference of potential. We 
 shall see later that conductors of the second class are always
 
 
 THE KLECTRIC CURRENT. 291 
 
 substances which act chemically upon the conductors of the 
 first class, and that a liquid like mercury, by which no such 
 chemical action is exerted, is in the first class. 
 
 185. The Voltaic Battery. Following out the indications 
 .of his theory, Volta undertook to increase the potential dif- 
 ference developed by contact, by bringing together a number 
 of successive pairs of the same metals. To do this he placed 
 a disk of paper or felt moistened with water, in which, to ren- 
 der it a better conductor, salt or acid was dissolved, between a 
 disk of copper and a disk of zinc. Such a set of disks we may 
 call an element. Having constructed a number of such ele- 
 ments, he placed one above the other, the zinc of the first 
 element being in contact with the copper of the second element, 
 so as to form a column, or, as it was called, a pile. This pile 
 was found to exhibit all the ordinary electric phenomena. It 
 behaved, as Volta said, like a battery of Leyden jars, with 
 this difference, that the jars, when discharged, need to be 
 charged from an outside source before another discharge can be 
 obtained from them, whereas the pile charges itself, so that 
 the electric effects can be obtained from it continuously. 
 
 The difference of potential between the copper wires joined 
 to the two ends of the pile is equal to the difference of poten- 
 tial due to the contact of copper and zinc, multiplied by the 
 number of elements in the pile. According to Volta's inter- 
 pretation of the facts, a difference of potential arises between 
 the first and second elements from the contact of the two 
 metals, and the potential of the two metals in the same ele- 
 ment is the same, by reason of the presence between them of 
 a conductor of the second class. The third element, in con- 
 tact with the second, introduces another difference of poten- 
 tial, so that the difference of potential between it and the 
 lower metal of the first element is double that due to one 
 element. Thus each successive element, and finally the copper 
 wire joined to the zinc of the last element, introduces an addi- 
 tional potential difference. 
 
 Volta soon recognized that a pile of this sort is not so well 
 adapted for continuous use as another arrangement of the
 
 292 THE ELECTRIC CURRENT. 
 
 conductors. He therefore constructed the arrangement which 
 he called the crown of cups, and which we now call the voltaic 
 battery. Each element of this battery is called a voltaic cell. 
 A typical voltaic cell consists of a glass vessel partly filled 
 with acidulated water, in which are immersed a plate of 
 copper and a plate of zinc. Wires are attached to the upper 
 ends of these plates, by which the difference of potential ex- 
 isting between them can be transferred to any desired point. 
 The voltaic battery consists of a number of such cells, of which 
 the zinc of one is joined by a wire to the copper of the next. 
 
 In the typical case of copper and zinc and of the cell 
 formed with them, it may be interesting to notice that when 
 zinc and copper are brought in contact, the potential of the 
 zinc becomes positive to that of the copper. The zinc is 
 therefore said to be electro-positive to copper. When the zinc 
 and copper of the cell are furnished with wires of the same 
 sort, the potential of the wire joined to the copper is positive 
 as compared with the potential of the wire joined to the 
 zinc. As we commonly think of the electric current as flowing 
 from the higher to lower potential in that part of the circuit 
 which lies outside the cell, we call the copper plate the posi- 
 tive pole of the cell, and the zinc plate the negative pole. 
 
 Volta considered the voltaic battery as a source of a con- 
 tinuous electric discharge. He found that his pile would give 
 the shock which is felt in the body when a Leyden jar is dis- 
 charged through it. He also found that when the terminal 
 poles of the pile, or of the battery, were joined by a thin wire, 
 the wire became heated. A similar heating effect had been 
 observed when a battery of Leyden jars was discharged 
 through a wire. By the use of modified forms of the battery, 
 a very great amount of heat was developed in the circuit, and 
 it was shown that the heat developed depends in some way 
 upon the nature of the materials composing the circuit. The 
 quantitative law of the development of heat was discovered by 
 Joule in 1842. 
 
 186. Chemical Action of the Current. When Volta'a ac- 
 count of the battery reached England in 1800, two members of .
 
 THE ELECTRIC CURRENT. 293 
 
 the Royal Society, Nicholson and Carlisle, constructed a bat- 
 tery and made experiments with it. In one of these they in- 
 troduced the ends of the terminal wires into a drop of water, 
 and noticed that bubbles of gas arose from them. They pro- 
 ceeded to investigate this phenomenon with an apparatus con- 
 structed as follows: Two test tubes filled with acidulated 
 water were placed with the open ends down in acidulated 
 water contained in a vessel, the water being held up in the 
 tubes by atmospheric pressure. In the open ends of the tubes 
 were placed small platinum plates, to which were joined the 
 terminal wires of the battery, these wires being covered with 
 insulating material, so that the current could not enter the 
 water except from the platinum plates. When the circuit, was 
 completed, gases were evolved at both the platinum terminals, 
 and collected in the upper ends of the tubes. These gases, 
 when examined, were found to be oxygen and hydrogen, the 
 constituents of water. It appeared from this experiment that 
 the electric current can decompose water into its constituents. 
 When the indications of this experiment were followed up, it 
 was found that very many compounds undergo a similar de- 
 composition. A common characteristic of all the compounds 
 which undergo decomposition was found to be that the com- 
 pound must be brought into the liquid state, either by solu- 
 tion in a solvent or by fusion. The products of decomposition 
 which are obtained are not always the constituents of the 
 compound which is dissolved. In the case just described, for 
 example, the products obtained are not the constitutents of the 
 acid dissolved in the water, but of the water itself. In such 
 cases the direct action of the current is supposed to be com- 
 plicated by secondary chemical actions. 
 
 The quantitative laws of the chemical action of the current 
 were discovered in 1834 by Faraday. 
 
 187. The Electric Arc. In 1800, by the use of a powerful 
 battery, Davy discovered that when the two terminal wires, 
 or better, when two pieces of charcoal or carbon which are 
 joined to the terminal wires, are first touched together, so 
 that the current is established, and are then slightly separated,
 
 294 THE ELECTRIC CURRENT. 
 
 the ends near the point of contact become intensely heated and 
 brilliantly luminous, and are connected by a column of what 
 looks like a flame, which is called the electric arc. The light 
 in the arc itself is generally bluish in color, and is less intense 
 than that from the carbon terminals. When carbon rods are 
 used as terminals, a small depression, called the crater, is 
 formed at the end of the positive terminal. The highest 
 temperature of the arc is obtained in this crater. It is esti- 
 mated to be as high as 3400 Centigrade. All known sub- 
 stances except carbon can be fused in the arc. 
 
 While the arc is established, both the carbon rods waste 
 away, the positive rod wasting twice as fast as the negative 
 rod. This effect is not due entirely to combustion, for the arc 
 may be established in a vacuum, and the wasting away of the 
 terminals occurs in that case also. 
 
 188. The Magnetic Field of the Current. In the year 1820, 
 the Danish physicist Oersted, who had been for some time 
 hoping to find a relation between the current and magnetism, 
 placed a wire carrying a current above a compass needle, and 
 parallel to it, and noticed that the needle turned out of the 
 magnetic meridian. When the current was reversed, the 
 needle turned out of the meridian in the other sense. When 
 the wire was placed below the needle, its deflection for the 
 same direction of the current was opposite to that obtained 
 when the wire was above it. 
 
 In the same year the action of the current on a magnet was 
 investigated quantitatively by Biot and Savart. In their ex- 
 periments the current traversed a long straight wire set up 
 vertically. By arranging magnets in the neighborhood so as 
 to neutralize the earth's magnetic field, a region was obtained 
 around the wire in which there was no perceptible magnetic 
 force, so long as there was no current in the wire. A small 
 magnet, suspended so as to turn with freedom in any direc- 
 tion, was used as an indicator of the effect of the current. 
 When the current was set up, this magnet assumed a position 
 in which it was tangent to a circle drawn around the wire as 
 centre, in a plane perpendicular to it. When the current was
 
 THK ELECTRIC CURRENT. 296 
 
 in one direction, the north pole of the magnet always pointed 
 in one sense around this circle, at whatever point on the cir- 
 cle it was placed. When the current was reversed, the direc- 
 tion in which the magnet pointed was also reversed. We may 
 express this result by saying that the current sets up and 
 maintains a magnetic field, in which the lines of force, in the 
 case of a long straight current, are circles having the current 
 as a common centre. 
 
 The direction of the lines of force depends upon the direc- 
 tion of the current. We consider the current as flowing from 
 higher to lower potential or from the positive pole of the 
 voltaic cell through the external circuit to the negative pole. 
 To express the relation between the direction of the current 
 and the direction of the lines of force, we may use several 
 modes of statement. Ampere's rule is that when an observer 
 swimming with the current looks toward the magnet, the north 
 pole of the magnet is deflected toward his left. Maxwell's 
 rule is that the direction of the current and the direction of 
 the lines of force are related as the translation and the rota- 
 tion of a right-handed screw. Another rule is that if the 
 right hand grasps the wire, with the thumb extended in the 
 direction of the current, the fingers encircle the wire in the 
 direction of the lines of force. 
 
 By allowing the indicator magnet to execute vibrations in 
 the magnetic field of the current at different distances from 
 the wire, Biot and Savart showed that the magnetic force in 
 the field at any point was inversely as the distance of that 
 point from the wire. Laplace showed that this law, which 
 expresses the integral effect of the whole straight current, 
 can be obtained from the hypothesis that each element of the 
 current acts on the magnet pole with a force which is inversely 
 as the square of the distance between the element and the pole, 
 and which is also proportional to the sine of the angle be- 
 tween the line joining the element with the pole and the direc- 
 tion of the current in the element. 
 
 The fact that a current acts on a magnet, by means of a 
 magnetic field which it sets up, suggested that two currents
 
 296 THE ELECTRIC CURRENT. 
 
 might perhaps interact with each other "by means of their mag- 
 netic fields. By experiments instituted to test this suggestion, 
 Ampere showed that in fact currents do interact with each 
 other. He furthermore determined a formula or law, ex- 
 pressing an action between any two elements of current, by 
 means of which all the forces exerted by currents on one an- 
 other can be calculated. 
 
 So long as we confine our attention to limited portions of 
 two circuits carrying currents, and consider only the forces 
 exerted by those portions on each other, we may express the 
 general mode of action discovered by Ampere by saying, that 
 parallel currents which are in the same sense attract each 
 other and those which are in opposite sensea repel each other. 
 If the currents are not parallel, the forces are such as would 
 be exerted by parallel components of the two currents, and are 
 therefore attractions if the components are in the same sense, 
 and repulsions if they are in opposite senses. 
 
 When we consider the action of two complete circuits on 
 each other, or of portions of two circuits so shaped as to be 
 practically equivalent to complete circuits, we are much 
 assisted by an analogy discovered by Ampere between the 
 magnetic field of a circuit and the magnetic field of a so- 
 called magnetic shell. The magnetic shell is an hypothetical 
 form of magnet, consisting of a thin sheet of steel whose 
 faces are uniformly magnetized positively and negatively re- 
 spectively. For convenience in representation and description 
 we shall consider only the plane magnetic shell. On considera- 
 tion of the field of force around such a shell, it is plain that 
 the lines of force which originate on its north or positive face 
 will be symmetrical curves passing around its outside edge 
 and terminating at points corresponding to the points of origin 
 on its south or negative face. When this system of lines of 
 force is compared with the lines of force in the magnetic field 
 of a current, travelling in a circuit which coincides with the 
 edge of the shell, the two systems are found to be identical 
 in form. The direction of the lines of force will also be the 
 same in both fields, if the current, to an observer looking at
 
 THE ELECTRIC CURRENT. 297 
 
 the south face of the shell, is travelling in the equivalent 
 circuit in the clockwise direction. Ampere's experiments 
 proved, in effect, that the forces between two closed circuits 
 are the same as those between the two magnetic shells which 
 are equivalent to them. Now the forces between two mag- 
 netic shells can be immediately perceived from the known 
 laws of magnetic action, and hence the forces between the 
 equivalent currents can be determined. 
 
 189. Ampere's Theory of Magnetism. Ampere used the 
 principles which he discovered as a basis for a theory of mag- 
 netism. In general he adopted a theory similar to that of 
 Weber, in so far as he supposed each molecule of iron to be a 
 separate and permanent magnet. He then explained the mag- 
 netic condition of the iron molecules by supposing that each 
 of them has an electric current continually circulating about 
 it. Such an electric current would set up a magnetic field 
 similar to that around a very small magnet, and so the mag- 
 netism of the molecule is accounted for as an electric phe- 
 nomenon. This theory of'magnetism, or one that is essentially 
 similar to it, though expressed in other terms, is still the 
 most satisfactory way of explaining natural magnetism. 
 
 190. Electromagnets. If a long wire is coiled into a tight 
 spiral, each turn is practically a small closed circuit and 
 acts like a magnetic shell, so that the magnetic field of the 
 whole spiral is like that of a pile of magnetic shells arranged 
 with their similar faces in the same direction. Such a pile 
 of magnetic shells is plainly equivalent to a magnetized bar, 
 and the magnetic field of such a bar is similar to that of the 
 spiral. The spiral used in this way to replace a magnet was 
 called by Faraday a solenoid. 
 
 An important difference between the field of the solenoid 
 and that of the magnet must be specially noticed. The lines 
 of force of a magnet, so far as they can be traced by experi- 
 ment, run from the north end of the magnet to its south end. 
 On the other hand, the lines of force of the solenoid are closed 
 curves, any one of which may be traced from a point at one 
 end of the solenoid through the region outside it to the other
 
 20R THK ELKCTRIC CURRKNT. 
 
 end, in which respect it is like a line of force of the magnet, 
 and then contimi'S in the mma sen^e within the solenoid to 
 the point of beginning. The lines of force of the solenoid 
 therefore form a bundle of closed curves. To see whether there 
 is any counterpart in the magnet to the lines of force within 
 the solenoid, it is necessary to measure the magnetic force 
 inside of the magnet. This can only be done by cutting cavi- 
 ties in the magnet at different points and by observing the 
 magnetic force within them. The theory shows that the mag- 
 netic force which will thus be observed will differ with the 
 shape of the cavity. If the cavity is always a narrow crevasse, 
 or disk-shaped cavity, with its faces perpendicular to the di- 
 rection of the force, the lines of force determined by observa- 
 tions within it will be similar to those within the solenoid. 
 The magnetic force determined in such a cavity is called the 
 magnetic induction, and the lines of force thus determined are 
 called linos of induction. We may therefore express the rela- 
 tion between the solenoid and a magnet by saying, that the 
 lines of force of the solenoid are similar to the lines of induc- 
 tion of the equivalent magnet. For a similar reason we may 
 say in general that the lines of force of a current are similar 
 to the lines of induction of the equivalent magnetic shell. 
 
 If a bar of soft iron is placed within the solenoid, while it 
 carries a current, the magnetic field of the solenoid will make 
 the bar a magnet. Such an arrangement is called an electro- 
 magnet. The magnetization of the bar disappears, or nearly 
 so, when the current in the solenoid ceases. Its. intensity of 
 magnetization depends upon the quality of the iron, but 
 mainly upon the strength of the magnetic field due to the 
 current, and this depends on the strength of the current and on 
 the number of turns made by the circuit about its axis in a 
 unit of length. 
 
 191. Galvanometers. Any instrument used to detect the 
 presence of a current, or to measure its strength by observa- 
 tions of the interaction between the current and a magnet, ia 
 called a galvanometer. A simple circuit of wire placed ver- 
 tically in the plane of the magnetic meridian, with a magnet
 
 THK KLKCTRir CURRENT. 299 
 
 suspended in the middle of it, will answer this purpose. The 
 portions of the current, flowing above the magnet in one di- 
 rection and below it in the. opposite direction, unite in turning 
 the magnet out of the magnetic meridian. The effect is natur- 
 ally heightened when the wire is turned on itself many times 
 into a flat spiral, the strength of the magnetic field increas- 
 ing with the number of turns. In one of the early forms of 
 galvanometer, constructed by Schweiger, a coil or circuit of 
 this sort was used in combination with an astatic needle. The 
 astatic needle, or system, consists of two similar light magnets 
 held rigidly parallel to each other by a short connecting rod. 
 These magnets are magnetized in opposite senses, and one of 
 them a little more strongly than the other. When such a pair 
 of magnets is suspended, the stronger one will overpower the 
 other, and its north pole will .point toward the north, but the 
 directive action of the pair is much feebler than that of either 
 one of them. In Schweiger's instrument the lower magnet of 
 the two hangs within the coil, while the other one is above 
 it. When the current passes, its magnetic field turns both 
 these magnets in the same sense, and since the directive action 
 of the magnets in the field of the earth is very slight, the 
 deviation of the system will be very great in comparison with 
 that which the same current would produce in a single mag- 
 net. Very feeble currents may therefore be detected by this 
 instrument. 
 
 A galvanometer which is of special practical and theoretical 
 importance is the tangent galvanometer, so called from one of 
 its characteristic properties. The coil of this instrument con- 
 sists of a number of turns of wire wound in a circle, so that 
 the thickness of the coil is very small in comparison with^he 
 radius of the circle. This circle is set up on edge in the 
 plane of the magnetic meridian, and a short magnet is sus- 
 pended at its centre. When a current is sent through the 
 coil, the magnet is turned out of the meridian, and assumes a 
 position in which the couple exerted on it by the earth's field 
 is in equilibrium with the couple exerted on it by the field 
 of the coil. The study of the field of the coil shows that it is
 
 300 THK ELECTRIC CURRENT. 
 
 of constant intensity in a region lying around the centre of 
 the coil, so that the force exerted by the field of the coil on the 
 magnet's pole does not change with the deviation of the mag- 
 net. If we represent the angle of deviation by 0. the horizon- 
 tal intensity of the earth's field by H, and the strength of the 
 field of the current by R, and remember that the lines of force 
 of the current are at right angles to the plane of the coil and 
 BO also at right angles to the magnetic meridian, it is evident 
 that the couple exerted on the magnet by the earth's field is 
 proportional to H f\n<f> and the couple exerted by the mag- 
 netic field of the current is proportional to R co0, so that 
 we have H sin0 = R cos0, and the ratio of R to H is equal 
 to the tangent of the deviation. It is for this reason that this 
 galvanometer is called the tangent galvanometer. If we make 
 the supposition that the strength of the current in the coil is 
 proportional to the strength of the magnetic field which it sets 
 up, we may compare different currents by comparing the tan- 
 gents of the deviations which they occasion. 
 
 192. Electromagnetic Rotations. A number of arrange- 
 ments were constructed by Faraday in 1821 by which the mag- 
 netic force acting between a current and a magnet was made 
 to produce a continuous rotation of a part of the apparatus. 
 In a typical one of these instruments two shallow circular 
 troughs are placed concentric with each other around an up- 
 right post, on the top of which is fixed a bearing. Two similar 
 magnets are placed vertically on opposite sides of the ap- 
 paratus, so that similar poles stand each between the troughs, 
 and a little above their level, while the other poles are con- 
 siderably below that level. A light wire frame, consisting 
 of a horizontal portion, of one dependent portion which reaches 
 the inner trough, and of two dependent portions which reach 
 the outer trough, is mounted on a rod whose end rests in the 
 bearing on the top of the post, so that the frame can turn 
 about the rod as a vertical axis. Mercury is poured into the 
 troughs until the ends of the dependent wires are always in 
 contact with it as the frame turns round its axis. Wires 
 from a voltaic battery introduce the current into one trough,
 
 THE ELECTRIC CURRENT. 301 
 
 from which it passes through the frame, and so out through 
 the other trough. When the current is introduced, the frame 
 maintains a continuous rotation about its axis. This move- 
 ment may be understood by considering the force between one 
 of the magnet poles and that part of the movable circuit which 
 is near it. If the circuit were at rest and the magnet pole 
 were free to move, it would be attracted toward one face of 
 the circuit, would pass through it, and then be repelled by 
 the other face. That is, the magnet pole would move as nearly 
 as possible along the lines of magnetic force of the current. 
 Since the magnet is fixed, it is the circuit which moves in the 
 opposite direction to that in which the magnet would move 
 in the case supposed. If the circuit were everywhere solid, it 
 would not be possible for the magnet pole to pass through its 
 plane without carrying its other pole with it, and the effect of 
 the two poles being opposite, no continuous rotation would be 
 set up. By making a part of the circuit fluid, we have ar- 
 ranged it so that the solid part of the circuit can pass from 
 one side to the other of one magnet pole without being affected 
 by the opposite pole. 
 
 The work which is done in sustaining these motions comes 
 from energy supplied to the circuit by the battery. As we 
 shall afterwards see more at length, the circuit when in motion 
 is not heated so much as it is when it is at rest. Ihe work 
 which is done during the motion is the equivalent of the heat 
 which has disappeared. 
 
 193. Electromagnetic Induction. The production of a mag- 
 netic field by a current suggested to many observers the pos- 
 sibility of the production of a current by means of a magnetic 
 field. For some years all endeavors to obtain such a current 
 were unsuccessful. In 1832 Faraday succeeded in obtaining 
 the expected action. In his first experiment Faraday used 
 two concentric spools of wire, one of which could be connected 
 with a voltaic battery. This he called the primary coil. The 
 other, called the secondary coil, was connected with a sensi- ' 
 tive galvanometer. With this arrangement Faraday found 
 that, when the current of the battery was thrown into the
 
 30^ THK ELBCTKIC CURRENT. 
 
 primary coil, the galvanometer was deflected so as to indicate 
 a current in the secondary con. This current in the secondary 
 coil was only temporary. While the current in the primary 
 coil was maintained unchanged, the deflection of the galvano- 
 meter ceased and no current appeared in the secondary coil. 
 When the circuit of the primary coil was broken, so that the 
 current in it ceased, a deflection in the opposite sense to the 
 one observed before occurred in the galvanometer, indicating 
 a current in the opposite sense in the secondary coil. This 
 current was of course also only temporary. The temporary 
 currents thus set up on making and breaking the primary cir- 
 cuit were called by Faraday induced currents, and they were 
 said to be produced by electromagnetic induction. By observa- 
 tions of the deflections of the galvanometer, Faraday proved 
 that the sense in which the induced current circulates in the 
 secondary coil is opposite to that of the current in the primary 
 coil when the primary circuit is made, and is the samo as that of 
 the current in the primary coil when the primary circuit is broken. 
 
 Reasoning that the setting up of a current in the primary 
 coil is, in effect, bringing the primary coil to its position 
 near the secondary from an infinite distance, Faraday next 
 tried the experiment of moving the primary coil, while the 
 current was established in it, up to the secondary coil. He 
 found then that the galvanometer was again deflected, showing 
 a current in the secondary coil, in the same sense as that of 
 the current produced when the current was thrown into the 
 primary coil in the first experiment. When the primary coil 
 was rapidly removed from the neighborhood of the secondary 
 coil, a current was produced in the secondary coil in the oppo- 
 site sense. 
 
 Faraday next substituted a magnet for the primary coil, 
 and found that, when it was moved up to the secondary in 
 the proper way, an induced current was developed, and that 
 an induced current in the opposite sense was developed when 
 the magnet was removed. The direction of the induced cur- 
 rents produced by the movement of a magnet may be readily
 
 T11K ELECTK1C CURRENT. 303 
 
 obtained from the rule already given, by remembering the 
 relation between the lines of force of a magnet and the direc- 
 tion of the current in the solenoid which is equivalent to the 
 magnet. 
 
 Faraday considered the induced current to result from the 
 change of the magnetic field around the circuit in which it 
 is produced. He looked on a magnet as a body which carries 
 with it a set of lines of magnetic force, and he described the 
 production of the induced current by the movement of a mag- 
 net as occurring whenever the circuit in which it arises cuts 
 through lines of force. It is perhaps easier for us to con- 
 sider the induced current as produced by an alteration in the 
 magnetic field enclosed by the circuit. The induced current is 
 produced only while the field is changing, and ceases when the 
 field becomes constant. 
 
 The principal phenomena of induced currents were dis- 
 covered by Joseph Henry about the same time that they were 
 discovered by Faraday and independently of him. 
 
 One day after a lecture Faraday had his attention called 
 by a gentleman in the audience to a fact which he had ob- 
 served, that when a circuit is broken in which a coil of wire 
 is contained, a bright spark appears at the break. On investi- 
 gating this phenomenon, Faraday found that the spark is 
 brighter when the wire in the circuit is in the form of a coil 
 than when the same wire is in the circuit, but is not coiled 
 up; and reflecting on this, he perceived that the spark is due 
 to an induced current set up in the circuit itself by the 
 change in its own current. The action to which this effect is 
 due is called self-induction. Self-induction was also discovered 
 and studied by Henry. If we apply the rules for the produc- 
 tion of the induced current to a single circuit, considering it 
 to act as both primary and secondary circuit at once, we see 
 that when a current is thrown into the circuit, an induced cur- 
 rent will be developed in the opposite sense. The effect of this 
 current is to retard the full development of the primary cur- 
 rent, so that the current will be established more slowly in a 
 circuit which is coiled up so that its parts can act inductively
 
 301 THE ELECTRIC CURRENT. 
 
 on one another, than it will be in the same circuit if it is not 
 coiled. When the circuit is broken, the departure of the cur- 
 rent produces an induced current in the same sense as that of 
 the original current, so that a momentary current of greater 
 strength is developed. It is to this current that the spark is 
 due which is observed when a circuit is broken. 
 
 A general rule, by which the direction of the induced cur- 
 rent in a circuit may be determined, was given by Lenz, and 
 is known as Lenz's law. If we examine the direction of the 
 induced current produced by bringing a primary current to- 
 ward the secondary circuit, we notice that it is in the opposite 
 direction to the current which would attract the primary 
 current. The induced current in this case therefore tends to 
 repel the primary current, or to oppose the work which is 
 being done on the primary circuit. In all other cases the 
 same general rule holds, and we may state Lenz's law by say- 
 ing, that the sense of the induced current is always such that 
 the force exerted by the current opposes the action by which 
 it is produced. 
 
 An interesting experiment which exhibits Lenz's law was 
 performed by Arago, several years before the induced current 
 was discovered. He arranged a copper disk so that it could be 
 rotated rapidly around a vertical axis, and suspended above it 
 a long magnet. When the disk was rotated the magnet was 
 deflected in the sense of the rotation, and when suspended so 
 as to be capable of free rotation around the axis of the disk, 
 it rotated in the same sense as that in .which the disk was 
 rotated. Arago could not explain this experiment, and it re- 
 mained unaccounted for until Faraday discovered induction. 
 It was then explained by Faraday as a consequence of the 
 induction of currents in the copper disk by its movement in 
 the magnetic field of the magnet. According to Lenz's law, 
 the currents thus induced are all in such a sense as to oppose 
 the motion producing them, and hence there arises a force 
 between the disk and the poles of the magnet, tending to pre- 
 vent the rotation of the disk, or what amounts to the same 
 thing, to set up a rotation of the magnet in the sense of the
 
 THK KLICTRIC CURBKXT. 805 
 
 rotation of the disk. That currents of the sort assumed in 
 this explanation really exist in the disk can be shown by 
 rotating a disk in a strong magnetic field, and touching two 
 parts of it with the terminals of wires leading to a galvano- 
 meter. With this arrangement a continuous current will pass 
 through the galvanometer so long as the disk is rotated. The 
 arrangement is therefore a machine for the production of the 
 electric current by motions in a magnetic field. The first 
 model of such a machine was constructed by Faraday. 
 
 The extensive use of the electric current in our modern 
 life originated in this discovery of Faraday's, by which it ia 
 made possible to use mechanical energy directly and on a 
 large scale, for the production of the current. The so-called 
 dynamo-machine, used for this purpose, consists generally of 
 a large fixed electromagnet, which furnishes a strong mag- 
 netic field, and of a rotating portion, called the armature, 
 made of a properly shaped mass of iron, over which wires are 
 wound in such a manner that as the armature is rapidly 
 rotated by an engine a current is developed in them. By 
 means of sliding contacts between these wires and the external 
 circuit, the current thus developed is led off from the ma- 
 chine. When the current is led into a similar machine the 
 interaction between the magnetic fields of its various parts 
 sets up and maintains a rotation of its armature. The ma- 
 chine thus used is called a motor. 
 
 Other forms of the dynamo-machine, which produce an 
 alternating instead of a direct current, and other motors 
 which can be operated by the alternating currents, have been 
 devised and are now very extensively used. 
 
 The induction coil, called also the Ruhmkorff coil, or the 
 inductorium, is an instrument used to produce induced cur- 
 rents with high electromotive forces. It consists of a cylin- 
 drical primary coil of coarse wire, wound around a central 
 core of iron, and a secondary coil, outside the primary and 
 concentric with it, containing very many turns of fine wire. 
 An automatic circuit breaker is employed, by which the cur- 
 rent from a battery is rapidly made and broken in the primary
 
 306 THE ELECTRIC CURRENT. 
 
 circuit. By the addition of a condenser to the primary cir- 
 cuit, which is charged when the circuit is made and is dis- 
 charged through it when it is broken, the development of the 
 primary current is delayed and its annihilation is hastened, 
 so that the electromotive force of the induced current in the 
 secondary coil, set up on making the primary circuit, is de- 
 pressed, and that of the current set up on breaking the 
 primary circuit is heightened. The consequence is that the, 
 induced current formed at the breaking of the primary is 
 able to leap over larger gaps than the other, and so, when the 
 current from the coil is sent through a non-conductor, like a 
 rarefied gas, it acts almost as if it were passing in one direc- 
 tion only. 
 
 104. Thermoelectricity. In our study of Volta's series it 
 was stated that no current will be set up in a circuit formed 
 exclusively of conductors of the first class, or that, in other 
 words, such a circuit will be in electric equilibrium. It was 
 discovered in 1822 by Seebeck that .this equilibrium is dis- 
 turbed when one of the junctions of the conductors composing 
 this circuit is heated. In that case a continuous current flows 
 in the circuit, so long as the difference of temperature between 
 the heated junction and the others continues. The current 
 thus formed is called the thermoelectric current. The strength 
 of the thermoelectric current depends on the difference of 
 temperature between the heated junction and the others. It 
 also depends upon the nature of the conductors at the heated 
 junction. These conductors are usually metals, and for each 
 pair of metals the current which is developed may be accounted 
 for by supposing that a difference of potential exists at the 
 point of contact between the metals, and that this difference of 
 potential changes with the temperature. This difference of 
 potential is not that discovered by Volta, but very much less 
 than that. The rate at which this difference of potential 
 changes with the temperature is called the thermoelectric 
 power of the pair of condurtors, or of the thermocouple or 
 element. That conductor of the couple from which the thermo- 
 electric current flows to the other, across the heated junction,
 
 THK ELECTRIC CURRENT. 307 
 
 is said to be thermoelectrically positive to the other conduc- 
 tor. Thus, for example, when a couple is formed of, antimony 
 and bismuth, it is found by experiment that the current flows 
 from the antimony to the bismuth across the heated junction; 
 the antimony is therefore thermoelectrically positive to the 
 bismuth. Experiment shows that all the metals can be ar- 
 ranged in a Beries, such that each successive member of the 
 series is thermoelectrically positive to those which tollow it, and 
 thermoelectrically negative to those which precede it. 
 
 The current produced in an ordinary circuit by a single 
 thermoelectric element is very slight, but a number of such 
 elements may be joined in succession, so that their differences 
 of potential are added to each other, and the current pro- 
 duced by them when their alternate junctions are heated may 
 be very considerable. - Such an arrangement of elements, called 
 a thermopile, is a very sensitive instrument for the detection 
 of radiant heat. 
 
 It was discovered by Peltier in 1834 that when a current 
 from a battery or any other source is passed through the junc- 
 tion of two metals in the sense of the current which would be 
 set up by heating that junction, the junction becomes cooler. 
 If the current is passed in the opposite sense, the junction is 
 heated. This effect is known as the Peltier effect. 
 
 195. Identity of Electricity from Different Sources. The 
 course of Volta's thought during the investigations which re- 
 sulted in his invention of the voltaic battery naturally led 
 him to believe that the effects produced by the battery were 
 electric effects, and that the battery was essentially equivalent 
 to a set of Leyden jars from which a continuous discharge 
 was passing. This supposed continuous discharge in the cir- 
 cuit of the battery was called the electric current. 
 
 The reasons which Volta had for the view which has been 
 described were, first, that a difference of potential could be 
 shown to exist between the two poles of the battery similar 
 in kind, though generally very much less in degree, to that 
 existing between the two coatings of a Leyden jar, and second, 
 that both the current and the discharge from the Leyden jar
 
 308 THK KLKCTRIC CURRENT. 
 
 heated the conductors through which they passed. It was felt 
 by Faraday that additional proof of this view was desirable. 
 He accordingly investigated the discharge of the Leyden jar, 
 in order to determine whether or not all the effects which 
 are produced by the current are also produced by it. He 
 found that, by proper arrangements, the discharge of the jar 
 could be made to produce chemical decomposition. Tke 
 amount of chemical action was so slight that it could only be 
 detected by him by a special artifice. He impregnated a piece 
 of blotting paper with a solution of starch and of iodide of 
 potassium, and sent the discharge through it between two ter- 
 minal wires whose ends touched the paper. When the dis- 
 charge had passed, small blue spots appeared around the ends 
 of the wires, due to the action upon the starch of the iodine 
 released by the chemical action of the discharge. The iodine 
 was evolved at both terminals, though more conspicuously at 
 one than at the other. This result could not be explained by 
 Faraday. We now know it to be due to the fact that the dis- 
 charge of the jar is oscillatory, or undergoes periodic changes 
 in direction. With our modern electric machines a continuous 
 chemical decomposition, as for example, of water, can be 
 maintained. The amount of this decomposition, even with the 
 most powerful machines, is extremely slight in comparison 
 with that produced by a battery. 
 
 Faraday also showed, by discharging Leyden jars through 
 a circuit containing a galvanometer, that the discharge 
 affected the magnetic needle, or produced a magnetic field. 
 The same thing was shown by Joseph Henry, who magnetized 
 sewing needles by placing them in the axis of a solenoid, and 
 sending the discharge through it. In the successive repeti- 
 tions of this experiment, Henry found that the magnetism of 
 the needles was not always the same, although the sense of 
 the discharge in the solenoid was always the same. The mag- 
 net was formed with its north pole sometimes at one end, 
 sometimes at the other. This remarkable result Henry ac- 
 counted for by supposing that the discharge of the jar was not 
 continuous, but oscillatory, the strength of the current 
 diminishing with each oscillation, and that the magnetic state
 
 THE ELECTRIC CURRENT. 309 
 
 of the needle was determined by the last oscillation which was 
 strong enough to reverse the magnetic condition impressed on 
 the needle by the oscillation before it. 
 
 From these experiments, and many others in which the 
 currents obtained from various sources were compared,, Fara- 
 day concluded that electricity from all these different sources 
 was identical in kind. 
 
 196. The Chemical Relations of the Current. We have 
 now passed in review the principal facts which were known 
 concerning the electric current up to the year 1834. It will 
 have been noticed that in most cases only qualitative state- 
 ments have been made about them. As the facts collected 
 became more numerous, and were shown by Faraday's ex- 
 periments, just referred to, to be intimately connected with 
 one another, the need of more precise quantitative knowledge 
 of the various relations of the current came to be felt. The 
 first domain in which such knowledge was supplied was that 
 in which the electric current is related to chemical action. 
 The investigation was carried out by Faraday, who was him- 
 self a chemist, as well as a physicist. 
 
 At the outset of the investigation, starting with the gen- \ 
 eral knowledge of the chemical action of the current which 
 has already been described, Faraday introduced a new set of 
 terms to describe the action under investigation. These 
 terms or names did not commit him to any theory of the 
 action. They are simply descriptive, and were so well fitted 
 to their purpose that they have always been employed since 
 they were introduced. In this nomenclature, the substance 
 which is decomposed by the current is called the electrolyte. 
 When decomposed it is said to be electrolyzed, and the pro- 
 cess of decomposition is called electrolysis. The two terminals 
 by which the current is introduced into the electrolyte are 
 called the electrodes, the positive one, or the one at higher 
 potential, being called the anode, the other the cathode. The 
 products of the electrolysis are called ions, the one which 
 appears at the anode being called the anion, the other the 
 cation. We shall use these terms freely in our discussion of 
 this subject.
 
 31U THE ELECTRIC CURRKNT. 
 
 Faraday devoted special attention to the electrolysis of 
 water. He had no means of measuring the strength of the 
 current which he used, but by altering various features of the 
 circuit, in such a way as not to change the current appre- 
 ciably, for instance, by changing the size of the electrodes and 
 by electrolyzing different solutions, he convinced himself that 
 the amount of electrolysis is independent of everything con- 
 nected with the circuit except the quantity of electricity 
 which passes. As we would now put it, the amount of electro- 
 lysis is proportional to the total current, and the rate at 
 which electrolysis takes place is proportional to the current- 
 strength. Faraday generalized this conclusion into the law 
 which we may state by saying, that the amount by weight of 
 an ion evolved at an electrode during the passage of a current 
 for unit time is proportional to the cur rent- strength. After 
 galvanometers were constructed by which an independent 
 measure of current strength was obtained, this law was in- 
 vestigated with the utmost care and completely verified. 
 
 Faraday also investigated the electrolysis of different com- 
 pounds, yielding different chemical elements as ions, by the 
 passage of the same current. He discovered by these experi- 
 ments a most important relation. To state it, it will be 
 necessary to say a word about the relations of the chemical 
 elements to each other. In making this statement we shall use 
 the ordinary atomic theory. 
 
 From the relations of the weights of chemical compounds 
 to each other, chemists have agreed that the chemical ele- 
 ments exist in the form of minute atoms, which combine with 
 one another to form chemical compounds; and that the atom 
 of any particular element has a definite weight, which is 
 Characteristic of that element. The weight of the hydrogen 
 Atom is usually taken as the standard or unit weight, and 
 the weight of the atom of another element expressed in terms 
 of that standard weight, is called its atomic weight. 
 
 When chemical compounds are examined it is found that 
 they generally consist of molecules containing a few atoms of 
 the elements which form the compound. In many instances 
 an element may be removed from such a compound, and an-
 
 THK KLECTRIC CURRENT. 311 
 
 other substituted for it. When this is done, it is often found 
 that the place vacated by one number of atoms of one element 
 is filled by a different number of atoms of the other element. 
 By such experiments, it may be determined how many atoms 
 of one element are equivalent in their combining power to a 
 single atom of another element. If we conceive such experi- 
 ments to be carried out by replacing hydrogen atoms by a 
 single atom of another element, the number of hydrogen atoms 
 displaced by the one atom of the other element is called the 
 valency of that element. Sometimes a small group of atoms 
 of different sorts, forming what is called a radical, will replace 
 a number of hydrogen atoms in a compound. In such a case, 
 the radical has a valency equal to the number of hydrogen 
 atoms which it displaces. 
 
 If we suppose an element having one valency to replace in 
 a compound another having another valency, it is plain that 
 the weights of the two elements which exchange places will be 
 proportional to their atomic weights divided by their valencies. 
 The ratio of the atomic weight to the valency is called the 
 chemical equivalent of an element. 
 
 We are now in a position to state the second law discovered 
 by Faraday, by saying, that the weights of the ions produced 
 at the various electrodes, when a current passes through dif- 
 ferent electrolytes, are proportional to the chemical equiva- 
 lents of the ions. To illustrate this law let us suppose that 
 the same current is used to electrolyze water, sulphate of 
 copper, and chloride of silver. The different elements con- 
 cerned, which we need to consider, are arranged in the follow- 
 ing table, with their respective atomic weights, valencies, and 
 chemical equivalents: 
 
 Atomic Chemical 
 
 Elements. Weight. Valency. Equivalent. 
 
 Hydrogen 1 1 
 
 Oxygen 16 2 8 
 
 Copper 63 2 31.5 
 
 Silver 108 1 108 
 
 Chlorine . . 35.5 1 35.5
 
 312 THE ELECTRIC CURRENT. 
 
 When the same current is sent through these compounds for 
 a while, and the products of electrolysis collected and weighed, 
 it will be found that, for every gramme of hydrogen, there 
 are obtained 8 grammes of oxygen, 31.5 grammes of copper, 
 108 grammes of silver, and 35.5 grammes of chlorine. 
 
 A conclusion can be drawn from the facts embodied in the 
 second law of electrolysis, which gives us an insight into the 
 nature of distribution of electricity. When we consider the 
 exact proportion between the current-strength and the weights 
 of the ions which are evolved at the electrodes, we are led to 
 consider the passage of the current from the electrolyte to the 
 electrode as effected by the transfer of electric charges to the 
 electrode by the ions which are developed on them. Each 
 elementary ion which has formed a part of a molecule of the 
 compound brings to the electrode a certain number of units of 
 valency. The second law shows that, for a given current, the 
 ions of different compounds bring to their respective elec- 
 trodes the same number of units of valency. If, therefore, we 
 suppose electricity to be divided into equal portions, which we 
 may call ionic charges and if we suppose that one such ionic 
 charge is associated with each unit of valency, and that the 
 charges of the ion are surrendered to the electrode when the 
 ion is developed on it, we then can account for both the laws of 
 electrolysis. It is plain that this conception is inconsistent 
 with any conception of electricity as being of the nature of a 
 continuous fluid. 
 
 197. Theories of Electrolysis. Within a few years after 
 the discovery of electrolysis, a theory was advanced by Grot- 
 thus to explain it. from which the more modern theories were 
 developed. Grotthus supposed that each molecule of the 
 electrolyte is made up of two equally and oppositely charged 
 parts, which parts become the ions when the molecule is broken 
 up. When a difference of potential exists between the 
 electrodes, so that there is an electric force in the electro- 
 lyte, these molecules arrange themselves in chains or rows, 
 with their positive charges all pointing in one direction, toward 
 the negatively charged electrode, or in the direction of the cur-
 
 THK ELECTRIC CURRENT. 313 
 
 rent. The molecules of the rows are then broken into their 
 ions by the electric force, and the positive ions move toward 
 the cathode, and the negative toward the anode, thus setting 
 free an ion at each end of the row, and bringing the remaining 
 positive and negative ions together so that they form new mole- 
 cule. These new molecules thus formed then arrange them- 
 selves as before under the electric force and the process is 
 repeated. This theory or description accounts for the evolution 
 of the ions at the electrodes and not in the body of the electro- 
 lyte, and it indicates a relation between the amount of electro- 
 lysis and the strength of the current. When Faraday's laws 
 were discovered, he found that they also could be explained by 
 Grotthus' theory, although Faraday himself preferred another 
 one, which was not, however, essentially different. 
 
 When electrolysis was studied more carefully, certain phe- 
 nomena were observed which could not easily be reconciled with 
 this theory. In particular, it was found that no matter how 
 small the difference of potential is between the two electrodes, 
 a current will pass between them through the electrolyte, and 
 electrolysis will occur. To account for this and for certain 
 other peculiarities observed, as, for example, the fact that the 
 conductivity of an electrolyte increases with the temperature, v 
 Williamson in England and Clausius in Germany modified the 
 theory of Grotthus. In their form of the theory use was made 
 of the hypothesis that the molecules of all bodies are in active 
 motion. This being admitted, it was assumed that occasionally, 
 when two molecules of the electrolyte encounter each other, 
 they are so affected by their mutual electric forces, or by the 
 shock of impact, as to be broken into their constituent ions. 
 These ions exist in the electrolyte for a time as free ions, and 
 during that time move toward the one electrode or the other, 
 according as they are positive or negative. This description of 
 electrolysis is no doubt a considerable advance on the one 
 which we first examined. 
 
 A further modification of the theory was rendered neces- 
 sary by a discovery of Kohlrausch. By determining the cur- 
 rents in different electrolytes under similar conditions, Kohl-
 
 314 THK ELKCTRIC CURRENT. 
 
 4 
 
 rausch showed that they can be accounted for by supposing 
 that the ions of each particular sort, when urged by a given 
 electric force, move through the electrolyte with a velocity 
 which is peculiar to that sort of ion, and which is the same 
 for that ion from whatever combination it is evolved. Thus, as 
 we may say, the conductivity of an electrolyte, that is, its 
 power of conveying current under standard conditions, is the 
 sum of two numbers characteristic of its ions. 
 
 Another experimental result was also influential in leading 
 to a modification of Clausius' theory. It was found that, dur- 
 ing the process of electrolysis, those parts of the electrolyte 
 near the electrode contain more than the average amounts of 
 the two ions. This movement of the ions toward their re- 
 spective electrodes was investigated by Hittorf. The relative 
 rates of movement for the different ions were found by Kohl- 
 rausch to agree with those which he had determined from his 
 observations of conductivity. 
 
 ' The requisite modification of the theory of electrolysis, to 
 account for these facts, was made by Arrhenius. The theory 
 which he developed is called the dissociation theory. Arrhe- 
 nius supposes that when a salt or any other substance is dis- 
 solved so as to form an electrolyte, a certain proportion 01 LS 
 molecules separate into their ions, so that a solution, whether 
 acted on by electric forces or not, contains a large number of 
 such free or dissociated ions. With each of these ions there 
 are associated electric charges of the proper sign, and propor- 
 tional in quantity to the valency of the ion. The effect of intro- 
 ducing an electric force in the electrolyte is to cause a drift of 
 these free ions toward the electrodes. When they reach the 
 electrodes, they give up their charges to them and are evolved 
 as the products of electrolysis. As the free ions are suc- 
 cessively removed in this way from the electrolyte, other mole- 
 cules of the solute are dissociated, so that the supply of free 
 ions is kept up. The fundamental assumption of this theory, 
 that a considerable proportion of the molecules of the solute 
 are dissociated in the solution into two parts, is borne out by 
 the effect which such solutes have in lowering the freezing 
 point and in raising the boiling point of their solutions.
 
 TlIK KLKCT.ilC CURRENT. 315 
 
 198. The Voltameter. Faraday suggested the use of the 
 electrolysis of water as a means of measuring the quantity of 
 electricity which passes through a circuit. The instrument 
 which he proposed he called the voltameter. In one of its 
 many forms, the voltameter consists of two graduated tubes, 
 which stand vertical and are connected at the bottom by a 
 short horizontal tube. To this connecting tube is joined an- 
 other through which the superfluous water may flow away 
 which is forced out as electrolysis proceeds. Platinum plates 
 are introduced into the two graduated tubes and joined to 
 wires carried through their walls. The apparatus, when ready 
 for use, is filled with acidulated water. When the current to 
 be measured is passed between the electrodes, electrolysis be- 
 gins and the two ions, oxygen and hydrogen, begin to collect 
 in the upper parts of the tubes. The quantity of electricity 
 which passes while the current is allowed to flow is then de- 
 termined by the amount of oxygen or hydrogen which is ob- 
 tained, and if the current is kept constant while it is flowing, 
 the amount of either gas evolved in unit time measures the 
 ^trength of the current. In the use of this instrument by 
 Faraday, the current indicated by it was measured in terms of 
 an arbitrary unit, namely, that current which will evolve a 
 determined quantity of gas in a fixed time. In the practical 
 use of the voltameter we still often employ such a unit, con- 
 sidering for example, that current to be unit current which 
 will evolve a cubic centimetre of hydrogen in one minute. As 
 we now, however, know the absolute value of the current which 
 will produce this result, we commonly express the strength of 
 a current in absolute units, even when we have used the volta- 
 meter to measure it. 
 
 It is obvious that any other electrolytic process may be 
 used as a means of measuring currents. In refined work it has 
 been found best to use the electrolysis of silver from a solution 
 of chloride of silver. The relation of the absolute value of the 
 current to the amount of silver which it will deposit from such 
 a solution has been very carefully determined. 
 
 199. The Voltaic Cell. Now that we have studied the pro- 
 cess of electrolysis, we are in position to examine the action
 
 816 THE ELECTRIC CURRKNT. 
 
 of the voltaic cell. As now becomes apparent, the conductor 
 of the second class which is interposed between the two plates 
 or poles of the cell, is an electrolyte. When the poles are 
 joined by a conductor other than this electrolyte, an unbal- 
 anced diii'erence of potential exists in the circuit, and a cur- 
 rent passes through the electrolyte. This current in the elec- 
 trolyte is from the electropositive element of the cell to the 
 electronegative element, that is, in the typical case already 
 described, from the zinc to the copper, and as it flows, the elec- 
 trolyte between these elements is broken into its ions. The 
 anion is evolved on the electropositive element, the cation on 
 the electronegative element. Now it always is the case, in any 
 voltaic cell, that the anions, which reach the electropositive 
 element, combine with its atoms to form molecules, which are 
 dissolved in the electrolyte. The electropositive element, which 
 is the negative pole of the battery, wastes away. The cation, 
 which is evolved at the other element, will sometimes interact 
 with it and sometimes not. If it does not, it appears as the 
 product of electrolysis at the positive pole. If it does interact 
 with that pole, some other substance is evolved by that action 
 as a product of electrolysis. 
 
 To illustrate this general description, we may consider the 
 simple voltaic cell formed of plates of copper and zinc, im- 
 mersed in a solution of sulphuric acid. When sulphuric acid 
 is dissolved, it dissociates into two positive hydrogen ions and 
 a negative ion, called the sulphion, containing one atom of 
 sulphur and four of oxygen. Each of the hydrogen ions is 
 univalent, and carries one positive ionic charge. The sulphion 
 is bivalent and carries two negative ionic charges. When the 
 circuit is joined, a difference of potential exists between the 
 zinc and the copper plates, the potential of the zinc being the 
 higher. The hydrogen ions move toward the copper plate, and 
 are evolved at it without acting on it chemically. The sul- 
 phions move toward the zinc plate, and when they reach it 
 combine each with an atom of zinc, so as to form sulphate of 
 zinc, which dissolves in the solution. Nothing appears at the 
 zinc plate as a product of electrolysis, and the plate gradually 
 wastes away.
 
 THE BLKCTBIO CU11RKNT. 317 
 
 In giving this description of the actions in the cell, the 
 order of events has been left as uncertain as possible. To make 
 a definite statement about it would involve deciding between 
 two rival theories of the action of the cell, known respectively 
 as the contact theory and the chemical theory. In the contact 
 theory, which was proposed by Volta, and which has been 
 highly developed by Lord Kelvin, the action of the cell is 
 ascribed to the unbalanced potential difference in the circuit, 
 arising from the contacts of the different parts of the circuit, 
 and the chemical action, by which the zinc is decomposed, re- 
 sults as a consequence of electrolysis set up by this potential 
 difference. In the chemical theory, held by Faraday, and advo- 
 cated by Lodge, the potential difference in the circuit is 
 ascribed to a tendency to chemical action between the zinc and- 
 the sulphions, by which a condition of strain is set up around 
 the zinc, and indeed throughout the cell. When the circuit is 
 joined, this strain can be relieved by the flow of electricity 
 around the circuit. Experiment has not yet decided between 
 these two theories. It is an argument in favor of the chemical 
 theory that it places the origin of the force which urges the 
 current round the circuit in the place where the energy of the 
 cell is being expended, that is. at the surface of contact be- 
 tween the zinc and the acid. On the contact theory, the largest 
 part of the potential difference by which the current is urged 
 around the circuit, arises at the contact between the zinc and 
 the wire by which it is joined to the copper, at a place, there- 
 fore, where no chemical action is going on, and where no 
 energy is being expended. Now in the cases of the production 
 of -the thermoelectric current and of the induced current, the 
 origin of the current is evidently in those parts of the circuit 
 in which energy is being expended, and the analogy of these 
 instances inclines us to favor the theory in which a similar 
 relation holds in the circuit of the voltaic cell. 
 
 When a cell of the sort just described is set in operation, 
 the current developed by it gradually diminishes in strength, 
 until it becomes much weaker than it was at first. This result 
 is ascribed to what we call the polarization of the cell. The
 
 318 THK ELECTRIC CURRENT. 
 
 hydrogen evolved at the copper plate collects on it in a very 
 thin layer, and so, in effect, partially replaces the copper plate 
 by a plate of hydrogen. Not only is this layer of hydrogen a 
 very poor conductor, but also the difference of "potential be- 
 tween it and zinc is less than that between copper and zinc. 
 For both these reasons the current of the cell is weakened. By 
 using platinum instead of copper, by roughening the surface of 
 the platinum, so that the hydrogen evolved on it more readily 
 forms bubbles and escapes, and by shaking the plate, much of 
 the hydrogen can be removed mechanically, and a cell obtained 
 in which the polarization is not great. Polarization may .be 
 avoided also by using other combinations of materials, so se- 
 lected that the ion which appears at the positive pole is of the 
 same sort as the pole itself. Thus in the Danlell cell, the ordi- 
 nary cell used in telegraphy, zinc and copper are used as the 
 elements, and two liquids are used, a solution of sulphate of 
 zinc around the zinc, and a solution of sulphate of copper 
 around the copper. These liquids are separated from each 
 other by a porous jar, or by the difference in their specific 
 gravities. When the current passes through this cell, it evolves 
 sulphions at the zinc, so that the zinc is reduced, and copper 
 ions on the copper, so that the copper plate merely becomes 
 thicker, but without the character of its surface changing. At 
 the surface where the two solutions meet, zinc ions proceeding 
 toward the copper plate meet with sulphions proceeding in the 
 opposite direction, and combine to form sulphate of zinc, so that 
 an additional effect of the action is to diminish the quantity 
 of sulphate of copper and increase the quantity of sulphate of 
 zinc. In order to prepare such a cell so that it will furnish 
 current for a long time without charging, crystals of sulphate 
 of copper are placed in the solution of sulphate of copper, so 
 that the strength of that solution may be kept up. 
 
 Other forms of cell are used, in which polarization is 
 avoided by surrounding the positive pole with some substance 
 which acts chemically upon the ions evolved, and removes them 
 from the pole. The Leclanche cell and the dry battery, used so 
 much for ringing electric bells and for other work of that
 
 THE ELECTRIC CURRENT. 319 
 
 kind, are examples of such cells. In them the depolarizing 
 material works slowly, so that the current cannot be run for 
 a long time without polarizing. But when the cell is used 
 only for a short time, it furnishes a strong current, and it ia 
 depolarized, by the time that it is again needed. 
 
 The storagfe >cell, developed from a cell constructed by 
 Plante, contains two lead plates, coattd with oxide of lead, 
 and immersed in sulphuric acid. When a current is passed 
 through this cell, a principal part of the action consists in a 
 reduction of the oxide on one plate to spongy lead, and the 
 conversion of that on ,the other plate into a higher oxide. 
 When the terminals of the cell are joined, after it has thus 
 been charged, the current which it delivers is accompanied by a 
 return of the surfaces of the plates to their initial condition. 
 
 200. Ohm's Law. In the first quarter century in which the 
 electric current was studied, the ideas concerning it lacked 
 definiteness, and the terms- by which it was described in differ- 
 ent cases were correspondingly vague. A general distinction 
 was drawn between intensity currents and quantity currents. 
 The intensity current was produced by such an arrangement 
 as the voltaic pile, or a voltaic battery containing a large num- 
 ber of cells in series, so that the difference of potential between 
 the two terminals, before the circuit was made, was high. It 
 was found that such a battery would send through a long wire 
 a current which did not differ very much from that which it 
 would develop in a shorter one. The quantity current was 
 developed by a single cell, or by a combination of cells so ar- 
 ranged as to be equivalent to a single one, in which the plates 
 were very large. Such a battery proved, by the amount of 
 electrolytic action which it would perform, that it was sending 
 out a large quantity of electricity, but it was not able to send 
 any considerable current through a long circuit. The reasons 
 for these differences and others like them were only vaguely 
 understood. 
 
 In 1826 G. S. Ohm published an account of experiments on 
 different circuits, in which the current was developed by a 
 thermoelement kept at a constant temperature. He measured,
 
 320 THE ELECTRIC CURRKNT. 
 
 by the action of the current on a magnet, the strengths of the 
 currents developed in his circuits as their lengths and other 
 characteristics were altered, and felt himself justified in an- 
 nouncing the law that the current strength is proportional to 
 a certain quantity characteristic of the thermoelement, or of 
 the source of current in the circuit, and inversely proportional 
 to the sum of two quantities, one of which is a constant for 
 any given source of current and the other varies with the re- 
 maining portion of the circuit. The sum of these two quanti- 
 ties Ohm called the resistance of the circuit. The quantity 
 characteristic of the source of current he called the electro- 
 motive force. In the following year, in the absence of exact 
 methods of measurement, by which this law could be really 
 tested, Ohm presented it as the result of speculation on the 
 nature of the electric current. It has since been abundantly 
 confirmed by the most careful observations. 
 
 Before proceeding to the discussion of Ohm's law we may 
 state it in terms which will afterwards be defined by saying, 
 that in any circuit the current equals the electromotive force 
 divided by the resistance. Of course it is understood, in 
 making this statement, that the quantities in the formula are 
 measured in a consistent system of units. It should also be 
 mentioned that exceptions to Ohm's law are found in certain 
 circuits in which the electric current is passing by means of 
 the electric discharge. 
 
 201. The Tangent Galvanometer. The tangent galvano- 
 meter has been briefly described in 191. It was invented by 
 Pouillet in 1837, and used by him in an investigation of Ohm's 
 law. In the use of the tangent galvanometer to measure cur- 
 rent, it is assumed that the current is proportional to the mag- 
 netic field set up by it. The strength of this field, and there- 
 fore the current strength, is proportional to the tangent of 
 the deflection. With this instrument very exact measures of 
 the current strength in different circuits may be made. 
 
 To illustrate the use of thfs instrument in examining Ohm's 
 law, let us suppose that we have a circuit in which, as the 
 source of current, there is placed a thermopile or a battery, so
 
 THE ELECTRIC CURRENT. 
 
 321 
 
 treated that the conditions to which the flow of current is due 
 remain constant. We then say that the electromotive force in 
 the circuit is constant. If the current from this source is led 
 through a tangent galvanometer and through an additional 
 conductor, called the external conductor, it will have a certain 
 value, indicated by the tangent of the deflection of the galvano- 
 meter needle. Suppose the external conductor a wire of a cer- 
 tain length. If we then lengthen the external conductor by 
 inserting a similar wire of the same length, the current ia 
 diminished. If we insert another similar wire, of the same 
 length, the current is diminished still further. Comparing the 
 strengths of the currents with each other, we find that they 
 may be represented by a formula, in which the numerator is 
 constant, and the denominator is the sum of two terms, one of 
 which is constant, and the other a constant multiplied by 1, 2, 
 or 3, according to the length of the wire in the circuit. It 
 seems obvious that the changes in the current strength are 
 due to the changes introduced in the circuit by the different 
 lengths of wire used. The effect of the wire seems to be pro- 
 portional to its length. We infer that the quantity, the value 
 of which changes as the length of wire changes, is character- 
 istic of the wire, and that the other quantity in the denomi- 
 nator, which remains constant, is a quantity similar in kind 
 to that related to the wire, and is itself related to the other 
 part of the circuit. We call the whole denominator the re- 
 sistance of the circuit, and distinguish the terms which refer 
 to the source of current and the external circuit respectively 
 as the internal and the external resistance. 
 
 If the external resistance is kept constant while the source 
 of current is changed, and if the difference of potential be- 
 tween the two ends of the external resistance is measured by 
 an electrometer, it is then found that the current in the ex- 
 ternal resistance is proportional to the difference of potential 
 between its ends. We shall subsequently see how this observa- 
 tion is generalized into the statement that the current is pro- 
 portional to the electromotive force.
 
 322 THK ELECTRIC CURRKNT. 
 
 In describing the use of the voltameter, it was stated that 
 currents are measured with it on the assumption that the cur- 
 rent strength is proportional to the rate at which electrolysis 
 goes on. Tn using the tangent galvanometer the current is 
 measured by assuming that it is proportional to the strength 
 of the magnetic field which it sets up. Experiment must be 
 employed to determine whether or not these two independent 
 modes of measuring current are consistent with each other. 
 This experiment consists in measuring currents in various cir- 
 cuits by both the voltameter and the tangent galvanometer. 
 It is found by such experiments that the two methods of meas- 
 uring are entirely consistent with each other, that is, the 
 magnetic force due to the current and the rate at which the 
 current performs electrolysis are exactly proportional to each 
 other. 
 
 202. Absolute Units. A very important step in advance 
 was taken by Weber in 1856. not only in electrical science, but 
 in the organization of physical science in general, by the in- 
 troduction of the absolute system of measurement. This abso- 
 lute system is one in which none of the units are entirely 
 arbitrary, except the three fundamental units of length, time, 
 and mass. All other units are developed from these by the aid 
 of suitable definitions based on physical laws. We may illus- 
 trate the definition of such a unit by referring to an example 
 which has already been given. The unit of force, the dyne, is 
 plainly defined in terms of these fundamental units. Now we 
 have defined the unit charge of electricity as that quantity 
 which will repel, in vacuum, an equal and similar quantity, at 
 the distance of one centimetre, with the force of one dyne. In 
 this definition the only quantities which enter are quantities 
 which are either the fundamental units themselves, or are de- 
 fined in terms of those units, and the unit quantity of 
 electricity is therefore definitely determined or defined in terms 
 of the fundamental units. The system of absolute units intro- 
 duced by Weber is one in which the various quantities which 
 occur in the subject of electricity, and which are capable of 
 measurement, are defined in an analogous manner in terms of
 
 THE ELECTRIC CURKKKT. 323 
 
 the fundamental units. This system has peculiar advantages, 
 not only in theoretical physics, but in the practical application 
 of electricity, in which it is of great importance to have simple 
 relations between the electric units and the units of force and 
 energy. We shall proceed to examine some of these units, and 
 illustrate in so doing the way in which their definitions are 
 derived. 
 
 203. Current. In defining the unit current, we proceed on 
 the hypothesis that current is a continuous transfer of 
 electricity. By the definition considered in the last section, we 
 have determined the unit quantity of electricity. We may 
 then, by its aid, define the unit current, as the current which 
 in unit time will transmit unit quantity of electricity through 
 any cross section of the conductor. We may interpret this 
 definition in terms of either the one-fluid or the two-fluid 
 theory of electricity, by considering negative electricity flow- 
 ing in one sense as equivalent to positive electricity flowing in 
 the opposite sense. The unit of current thus defined is called 
 the electrostatic unit of current. 
 
 Another absolute unit of current may be defined from the 
 magnetic relations of the current. By the aid of the unit 
 magnet pole, which has already been defined, we may measure 
 the strength of a magnetic field, and so define the magnetic 
 field of unit strength, as the field in which a force of one dyne 
 will be exerted on a unit pole. Now a current in the circum- 
 ference of a circle will exert a magnetic force on a unit pole 
 at its centre which is proportional to the circumference of the 
 circle, and inversely proportional to the square of its radius, 
 or, since the circumference of the circle equals 2wr, is propor- 
 tional to . We may therefore define the unit current as 
 
 the current which will set up at the centre of a circle of unit 
 radius, in which it is flowing, a magnetic field whose strength 
 is 2rr. A more usual definition .f unit current is obtained 
 from the analogy between the circuit and the magnetic shell, 
 the two definitions being, of course, consistent with each other. 
 The current thus defined is called the electromagnetic unit of 
 current.
 
 324 THE ELECTRIC CURRENT. 
 
 The fundamental units enter into the definitions of the 
 electrostatic unit and of the electromagnetic unit in different 
 ways, s(b that when the units are compared with each other by 
 measuring one of them in terms of the other, the ratio between 
 them is not a simple number, but is a number expressing a 
 velocity. This velocity is of the greatest theoretical import- 
 ance in the theories of electricity and optics. It was first de- 
 termined by Weber and Kohlrausch. In the method which 
 they used, a condenser of known dimensions was charged to a 
 known potential, so that the charge contnined by it was known 
 in electrostatic units. By discharging it through a galvano- 
 meter, the total current which passed through the galvano- 
 meter, or the quantity of electricity which passed, was meas- 
 ured in electromagnetic units. Expressed as a velocity, uie 
 ratio between the two units of quantity was found to be 
 311,000.000 metres per second, the electromagnetic unit being 
 the larger one. Many determinations of this velocity have 
 since been made, by methods which involve the comparison of 
 the units of various electrical quantities. The best and most 
 recent results agree in assigning to this velocity the value of 
 three hundred million metres per second. 
 
 204. Electromotive Force. If we still retain the hypothesis 
 that the, current is the transfer of electricity, and consider the 
 current flowing in a part of the circuit, the ends of which are 
 at the potentials V and V in electrostatic units, and if in the 
 time t the quantity of electricity Q in electrostatic units is 
 transferred from one end of the conductor to the other, the 
 work which is done is equal to Q (V-V), and the rate at which 
 work is done is equal to this quantity divided by t. Now the 
 quantity of electricity transferred, divided by the time, meas- 
 ures the current strength / in electrostatic units. The rate at 
 which work is being done, or energy is being expended, in this 
 part of the circuit is therefore equal to I (V-V). 
 
 In a complete circuit in which a current exists, energy is 
 being expended at a certain rate. We assume that there exists 
 in the circuit what we call an electromotive force, which is 
 analogous to the difference of potential of the last paragraph,
 
 THB ELECTRIC CTJBBINT. 326 
 
 in thut it conforms to an equation of the same form. We 
 therefore define the electromotive force as the power of main- 
 taining the expenditure of energy in the circuit, and express 
 its relation to the energy and the current by the formula 
 W=IE, in which IF represents the rate at which energy is be- 
 ing expended in the circuit, and E the electromotive force. 
 The unit electromotive force in the electrostatic system is 
 therefore the electromotive force in a circuit in which the 
 electrostatic unit of current will expend one erg in one second. 
 
 We measure electromotive force in the electromagnetic sys- 
 tem by the same relation. That is, the energy expended in a 
 circuit is defined to be equal to the product of the current and 
 the electromotive force measured in electromagnetic units. 
 The unit electromotive force in the electromagnetic system is 
 therefore the electromotive force in a circuit in which the 
 electromagnetic unit of current will expend one erg in one 
 second. 
 
 205. Resistance. In 200 we have seen that we may set the 
 current in a circuit proportional to the electromotive force 
 divided by a quantity which we have called resistance, and 
 which is characteristic of the circuit. We may measure re- 
 sistance in the absolute system by means of the relation ex- 
 pressed in the equation IRE, from which, since we already 
 know how to measure the current 7 and the electromotive force 
 E in absolute units, we may measure the resistance R in abso- 
 lute units. In the equation as written, the quantities are ex- 
 pressed in the electrostatic system, and from it we see that we 
 may define the electrostatic unit of resistance as the resist- 
 ance of a circuit in which the electrostatic unit of electro- 
 motive force maintains an electrostatic unit of current. Simi- 
 larly the electromagnetic unit of resistance is the resistance 
 of a circuit in which the electromagnetic unit of electromotive 
 force maintains an electromagnetic unit of current. 
 
 In connection with the subject of resistance it may be 
 stated that the resistance of a cylindrical conductor like a 
 metallic wire, is found to be proportional to the length of the 
 wire and inversely proportional to its cross- section. It is also
 
 326 THE ELECTRIC CURRENT. 
 
 proportional to the specific resistance of the substance, that is, 
 the resistance of a cylinder of the same substance of unit 
 length and of unit cross section. 
 
 206. Joule's Law. The energy which is introduced into the 
 circuit from the source of the current is expended in tne cir- 
 cuit. In a homogeneous part of the circuit, as, for example, in 
 a wire joining the poles of a battery, this energy is converted 
 into heat. In 1842 Joule carried out a series of experiments 
 to discover the relation between the heat developed in such a 
 conductor and the current in it. He found that for a given 
 conductor the heat developed is proportional to the square of 
 the current, and that, for a constant current, the heat devel- 
 oped in different conductors is proportional to the resistance. 
 If we measure current and resistance in I he units already de- 
 fined, and measure heat in units of energy, we may show from 
 the equations of the last two sections that the heat developed 
 in unit time is equal to the square of the current multiplied 
 by the resistance. For, the heat developed equals the energy 
 expended, and this, from $204, equals I(V-V'). From Ohm's 
 law applied to this part of the circuit we have IR=V-V\ 
 From these equations we obtain the relation which has been 
 stated, and which is known as Joule's law. 
 
 207. Helmholtz's Theorem. When we consider the whole 
 circuit, it generally happens that some of the energy expenued 
 in it is used in doing something else than heating the con- 
 ductor. It may, for example, do chemical work in electrolysis, 
 or heat the junction of two metals, or lift a magnet into a coil. 
 In all such cases as these, it has been found by experiment 
 that the rate at which energy is expended is exactly propor- 
 tional to the current. We may therefore express the relation 
 between the energy expended by the* source of the current and 
 the work that is done throughout the circuit by the equation 
 IE = 1 2 R+1A, in which the factor A expresses the rate at 
 which this extra work is done by a unit of current. From 
 
 this equation we obtain 7= p as the expression for the cur- 
 rent in the circuit. The numerator E-A. is obviously an
 
 THK ELECTRIC CURRENT. 327 
 
 electromotive force, and the equation shows that when work 
 is done in the circuit, in any other way than in heating the 
 circuit, the electromotive force in the circuit is less than that 
 introduced by the source of the current, by an amount wuiua 
 depends on the rate at which the extra work is being done. If 
 the electromotive force E is suppressed, leaving the conditions 
 in the circuit otherwise the same, the electromotive force A 
 will exist in the circuit in the opposite sense to that of E, and 
 a current will also exist in the circuit in the sense of the 
 electromotive force A so long as the conditions are maintained 
 to which that electromotive force is due. 
 
 We call the electromotive force A, which is developed by 
 doing extra or local work in a part of the circuit, the counter- 
 electromotive force. By this theorem we can explain the vari- 
 ous modes of producing the electric current. For example, when 
 the current moves a magnet, a counter-electromotive force is 
 set up in the circuit. A similar movement of the magnet, due 
 to any outside action, will cause the same electromotive force 
 to arise and so will produce an induced current. This relation 
 was first proved by Helmholtz, who used it in illustration of 
 the principle of the conservation of energy, for it follows, as 
 we have seen, as a consequence of that principle. Similarly, 
 when the current decomposes an electrolyte, and liberates dis- 
 similar ions, a counter-electromotive force is set up. If these 
 liberated ions are otherwise introduced into the electrolyte, the 
 same electromotive force will arise. Such a combination is a 
 voltaic cell. So also, if a current does work by heating the 
 junction of two metals, a counter-electromotive force is set up. 
 If the same junction is heated from an outside source, the same 
 electromotive force is set up, and we have the thermoelectric 
 current. 
 
 We may illustrate this theorem of Helmholtz by an experi- 
 ment of Joule's. Joule determined the amount of heat which 
 was developed when a certain mass of zinc was directly dis- 
 solved in sulphuric acid. He then determined the amount of 
 heat developed in the circuit of a battery, while the same 
 amount of zinc was consumed in the battery. He found that
 
 328 THE ELECTRIC CURRENT. 
 
 the same amount of heat was developed, with this difference, 
 that it appeared throughout the circuit instead of appearing 
 immediately at the place where the chemical action was going 
 on. He then inserted a magnetic motor in the circuit, which 
 was run by the current, and could be made to do work by 
 lifting a weight. So long as the motor was at rest, or was 
 running without doing work, the amount of heat developed in 
 the circuit, for the consumption of the same amount of zinc, 
 was the same as before. When the motor did work, less heat 
 was developed. The amount of heat which did not appear was 
 proportional to the work done by the motor, and measured in 
 terms of energy units was equal to it. 
 
 208. Practical Units. The direct measurement of current, 
 electromotive force, and resistance, in their absolute units, is 
 extremely difficult and in ordinary circumstances impracti- 
 cable. It has been found necessary to establish intermediate 
 standards, which are more easily used in ordinary measure- 
 ments, and to determine the values of those standards, once 
 for all, in absolute units. 
 
 If we know the value of the horizontal intensity of the 
 earth's magnetism, we can compare with it the magnetic force 
 set up by a current in the tangent galvanometer, and can cal- 
 culate the value of that current in electromagnetic units, if we 
 know the dimensions of the galvanometer. The absolute value 
 of a current is often determined in this way in ordinary labora- 
 tory practice, but unless the galvanometer is very exactly 
 made, and the operation with it conducted with extreme care, 
 the value obtained cannot be depended on as accurate. Accord- 
 ingly, several observers have compared the strength of a cur- 
 rent, measured with a very exact tangent galvanometer, with 
 the amount of silver which the same current deposits in a 
 second. They thus found that the amount of silver deposited 
 in one second by the electromagnetic unit of current is 11.175 
 milligrams. By the aid of this number a current which is 
 measured directly by the silver which it deposits can be ex- 
 pressed in absolute units.
 
 THK ELECTRIC CUKRENT. 329 
 
 The electromagnetic unit of current is not the one which is 
 used in ordinary practice. The practical unit of current, called 
 the ampere, is equal to lO" 1 absolute units of current. 
 
 To determine the absolute value of a resistance we must 
 determine in absolute units the current and the electromotive 
 force in the circuit. The method employed by the Committee 
 of the British Association on Electrical Standards will serve ?s 
 an illustration of how this may be done. The wire whose re- 
 sistance was to be determined was wound on the circumference 
 of a large circle into a coil whose ends were joined to each 
 other. This coil was mounted so that it could rotate at a de- 
 terminate rate around its vertical diameter. A small magnetic 
 needle was hung at the centre of the coil. When the coil was 
 turned, it cut through the lines of force of the earth's magnetic 
 field, and so set up induced currents in itself. The electro- 
 motive force in the coil was calculated from the rate of change 
 of the number of lines of force encircled by the coil. The in- 
 duced currents in the coil all tended to turn the magnet in the 
 same sense, and the strength of the magnetic field produced by 
 them, and so the strength of the current in the coil, was de- 
 termined from the deflection of the magnet. When the electro- 
 motive force and the current were both known in absolute 
 units, the ratio between them gave the resistance of the coil in 
 absolute units. By comparison with the resistance of this coil, 
 a wire of known resistance was constructed as a standard, and 
 by comparison with it standard sets of resistances have been 
 made, with which other resistances can be compared. 
 
 The electromagnetic unit of resistance thus determined is 
 too small to be of practical use. Instead of it a practical unit 
 is used, called the ohm, which was originally designed to be 
 equal to TO 9 electromagnetic units of ivsisUinco. It b;i- l>een 
 found more convenient to define the ohm as the resistance of a 
 column of mercury, one millimetre in cross-section and 106.3 
 centimetres in length, at the temperature of melting ice. The 
 resistance of such a column is very nearly equal to the ohm as 
 previously defined.
 
 330 THE ELECTRIC CURRENT. 
 
 The electromotive force of a circuit may be measured by 
 measuring the current and the resistance of the circuit in abso- 
 lute units. Certain voltaic cells have been constructed whose 
 electromotive force is very constant, and reproduced with great 
 precision when the cells are made up according to a prescribed 
 formula. The electromotive forces of these cells have been 
 very carefully determined, so that they serve as intermediate 
 standards. 
 
 The electromagnetic unit of electromotive force is too small 
 to be of use in practice. We use instead of it a practical unit, 
 called the volt, which is equal to 10 8 electromngnetic units of 
 electromotive force. The electromotive force of the Daniell's 
 cell is a little greater than one volt. 
 
 The energy expended in a circuit in one second is measured 
 by the product of the current and the electromotive force in 
 the circuit. When the current is the ampere and the electro- 
 motive force is the volt, me energy expended in one second, or 
 the rate at which energy is expended, is taken as a unit rate of 
 expenditure of energy. This unit is called the watt. It is 
 equal t<> TO 7 ergs per second. 
 
 209. Theories of the Electric Current. From the first the 
 electric current was thought of as a continuous transfer of 
 electricity around the circuit. In the one-fluid theory this 
 transfer was all in one direction; in the two-fluid theory posi- 
 tive electricity was supposed to move in one direction, and an 
 equal amount of negative electricity in the opposite direction. 
 It was plain that the interactions of currents could not be 
 accounted for by the known laws of electric attraction and 
 repulsion, but no one, not even Ampere, ventured to suggest 
 any modification of those laws by which the actions of cur- 
 rents could be explained. The question was finally taken up 
 by Weber, who undertook to explain the actions of currents 
 by the hypothesis that moving charges act on each other, not 
 only with their electrostatic forces, but also with other forces 
 which depend upon their velocities. By developing this 
 hypothesis, Weber was able to account for the actions of steady 
 currents on each other. By making the further hypothesis
 
 THK ELECTRIC CURRENT. 331 
 
 that additional forces arise when the velocities of the moving 
 charges are changing, Weber was also able to account for the 
 induction of currents. 
 
 The theory of Weber assumed that the actions between 
 currents take place in some direct manner, which does 
 not depend in any way on the medium between the currents, 
 and which depends only on the distance between them. It may 
 therefore be called a theory of electric action at a distance. 
 The researches in electricity and magnetism in which Faraday 
 was for many years engaged convinced him that electric and 
 magnetic actions are not simply actions at a distance, but take 
 place by means of the intermediate actions of some medium. 
 By reflecting on these views of Faraday, Maxwell was led to 
 attempt to give them mathematical form, and so to construct a 
 theory of what we may call medium-action. One form of his 
 theory, in which he describes a medium which will account for 
 electrostatic action, has already been described. One feature 
 of this theory, and indeed of Maxwell's theory in all its forms, 
 is that there can be no such thing as an open circuit, that is, 
 there can be no flow of electricity which begins and ends in a 
 limited conductor. According to his theory, the flow of elec- 
 tricity in the conductor is accompanied by an electric displace- 
 ment in the dielectric around the conductor, so that the cir- 
 cuit is completed through the dielectric. While this displace- 
 ment is being set up in the dielectric, it is so far like a cur- 
 rent that it produces the same magnetic field that a current 
 of the same strength and similarly distributed would produce. 
 
 In the most general form of Maxwell's theory, the hypo- 
 thesis is made that the ether and the electricity, which is dis- 
 tributed everywhere in it and which can be displaced in a di- 
 electric and move freely in a conductor, conform to the general 
 principles of mechanics, in so far that if, to the quantities 
 which enter into the general equations of dynamics, there be 
 given an electric interpretation, those equations will represent 
 the mode of action of the electric and magnetic fields. By the 
 development of this hypothesis Maxwell was able to show that 
 it leads first to the induction of currents, then to the actions
 
 dbli THE ELECTRIC CUKRENT. 
 
 of steady currents, and finally to the actions of electricity in 
 equilibrium. 
 
 If we adopt Maxwell's theory, we conceive the act of set- 
 ting up a current to involve not only starting a stream of 
 electricity along the conductor, but also setting in action we 
 might by analogy say, in motion the etherial mechanism in 
 the conductor and in the dielectric surrounding the conductor. 
 
 When this mechanism is in action, the region occupied by 
 it becomes the magnetic field of the conductor. To set up this 
 field requires energy, and since the energy from the source is 
 supplied only at a certain rate, and is at first divided be- 
 tween that expended in the circuit and that spent in setting 
 up the magnetic field, the current in the conductor rises only 
 gradually to its full value. When it is fully established the 
 maintenance of the action in the magnetic field requires no 
 more energy. According to the mode of representation adopted 
 by Poynting, the flow of energy from the source into the di- 
 electric does not stop when this steady state is reached, but 
 is continued at a constant rate, and the energy, passing 
 through the dielectric in a certain definite manner, reaches 
 the conductor by way of the dielectric and is transformed in it. 
 When the circuit is broken, and the flow of electricity ceases, 
 the conditions which sustain the action of the mechanism 
 cease also, and the energy stored up in the dielectric leaves 
 it and appears in the conductor as the energy of the extra 
 current. 
 
 When the current is steady, Poynting's theorem shows that 
 tho expenditure of oneriry in an ordinary homogeneous part of 
 the circuit, resulting in the development of hent in it, will be 
 uniform across the whole cross-section of the conductor. But 
 this will not be the case when the current varies periodically. 
 A current which so varies may be developed by a properly 
 constructed dynamo-machine, which is arranged to set up an 
 electromotive force in the circuit varying continuously and 
 periodically between two extreme and oppositely directed 
 values. Such a current is called an alternating current. It 
 was shown by Heaviside that the energy which enters the con-
 
 THE ELECTRIC CURRENT. 333 
 
 ductor from the dielectric, when the current in it is alter- 
 nating, is transformed into heat before the current distributes 
 itself uniformly over the whole cross-section of the conductor, 
 so that the principal development of heat is in the outer 
 layers of the conductor. This effect has been demonstrated by 
 experiment. The extent to which the current penetrates the 
 conductor depends on the period of the alternations, being 
 greater as this is longer. For very rapid alternations, the cur- 
 rent is confined almost entirely to the surface of the conductor. 
 
 This action of alternating currents is a consequence of 
 Maxwell's theory, and may be considered a verification of it. 
 
 Still, Maxwell's theory, if it had gone no further than this, 
 would have been little more than a simple alternative to the 
 theory of Weber, to be adopted by those who prefer to think 
 in terms of medium-action rather than in terms of action at a 
 distance. 
 
 Maxwell, however, was able to draw from this theory an- 
 other more important series of consequences, which do not fol- 
 low from Weber's theory, and which were subsequently veri 
 fied by experiment. We shall now turn our attention to this 
 portion of Maxwell's theory and to its experimental demon- 
 stration. 
 
 210. Electromagnetic Waves. It has already been de- 
 scribed how the flow of electricity in a limited conductor, on 
 Maxwell's theory, rs accompanied by an electric displacement 
 in the dielectric. If this flow alternates in direction, or ia 
 oscillatory, the displacement in the dielectric will undergo 
 similar oscillations. It follows from Maxwell's theory that 
 when such oscillations are set up, they will proceed outward 
 from their origin, as electromagnetic waves, with a definite 
 velocity. In the ether, according to the theory, this velocity is 
 that which expresses the ratio between the electrostatic and 
 the electromagnetic units. In other bodies it is equal to this 
 velocity divided by the square roots of their dielectric con- 
 stants. In all their essential characteristics these waves are 
 exactly like light waves. In default of direct experimental 
 evidence, Maxwell ventured to assume that the^waves of light
 
 334 THK ELtCTKIC CL'HKENT. 
 
 are electromagnetic waves, and to examine their properties for 
 a confirmation of his theory. A comparison of the velocity 
 which is the ratio between the two systems of units with the 
 velocity of light showed that the two velocities are very nearly 
 equal. More accurate determinations of both these quantities 
 have shown that they are really equal within the limits of 
 probable error. The velocity of light in any other body than 
 ether is equal to its velocity in the ether divided by its index 
 of refraction. If therefore light waves are electromagnetic 
 waves, the index of refraction of a substance ought to be equal 
 to the square root of its dielectric constant. In the case of 
 paraffine, which was the only body for which he knew both the 
 index of refraction and the dielectric constant, Maxwell found 
 that this relation nearly, though not exactly, held true. Later 
 investigation has shown that it rarely holds true with light 
 waves, but an explanation of this circumstance can be given 
 which permits us still to regard light waves as electromagnetic 
 waves. In addition, according to the theory, good conductors 
 ought to be opaque to light and poor conductors transparent, 
 for when the electric disturbance in the wave falls upon a 
 good conductor, it should set up a current and its energy 
 should be transformed into heat, while in a poor conductor it 
 should continue without any such transformation. Maxwell 
 found, in fact, that this relation holds generally true, although 
 the degree of transparency exhibited by the conducting metals 
 when in thin sheets was greater than he had expected. Later 
 observation, and an extension of the theory, have shown that 
 the facts of observation confirm the theory. 
 
 Maxwell was able to go no further than this, and the 
 electromagnetic theory of light, as it was called, remained an 
 interesting speculation, with some evidence in its favor, umtil 
 the year 1888, when the study by Hertz of electromagnetic 
 waves and their properties confirmed the general theory of 
 Maxwell in the most complete manner. 
 
 To set up electromagnetic waves, Hertz used the electric 
 spark passing between two knobs joined by short rods to two 
 similar metals plates standing in the same plane. This ar-
 
 THhi KLKCTKIC C'UKKKXT. 33-3 
 
 rangement is called the vibrator. As was shown by the ex- 
 periments of Faraday and Henry, and confirmed by the cal- 
 culations of Lord Kelvin and the observations of Feddersen, 
 the discharge in such an electric spark is oscillatory. It is not 
 the passage of electricity in one single leap across the gap, 
 but is instead a succession of passages of electricity, alter- 
 nately in opposite directions and in diminishing quantity, 
 until equilibrium is attained. These alternate passages occur 
 at equal intervals of time, and are therefore adapted to serve 
 as the origin of a short train of waves. 
 
 Hertz detected the passage of these waves by using a wire 
 rectangle or circle, in which a small gap was opened so that, 
 if a current was set up in the circuit, its presence could be 
 detected by a spark at the gap. He constructed this circuit of 
 such a length that the period of the free oscillation of elec- 
 tricity in it was the same as that of the discharge. The 
 electric impulses coming to it in the wave were therefore syn- 
 chronous with its own free vibration, and thus the electric 
 effects in it were heightened. This instrument was called the 
 resonator. After showing that he could detect an electric dis- 
 turbance, which was presumably an electric wave, whenever the 
 spark passed at the vibrator, Hertz used the resonator to com- 
 pare the velocity of electric waves in a wire with their velocity 
 in air. The results of these experiments were Inconsistent with 
 Maxwell's theory, but Hertz afterwards recognized that there 
 was probably some undiscovered error in the way in which the 
 experiments were made, and the observations of others who 
 have tried similar experiments confirm the conclusions of the 
 theory, that the velocity of the electromagnetic waves In the 
 wire is the same as that in the air. 
 
 According to the theory an electromagnetic wave is re- 
 flected with change of sign at a conducting surface on which 
 it falls perpendicularly. A node should therefore be developed 
 at that surface, and, if the reflected wave is sufficiently in- 
 tense, a standing wave, or a succession of nodes and ventral 
 segments, ought to be set up. By using a large sheet of zinc 
 as a reflector Hertz was able to detect several of these nodes,
 
 336 THE ELECTRIC CURRENT. 
 
 and so to measure the length of the electromagnetic wave. 
 Knowing the period of the wave by calculating it from the 
 dimensions of the vibrator, he was able to determine its 
 velocity. He found it to be of the same order of magnitude as 
 the velocity of light. Subsequent observations have proved 
 that the velocity of light and the velocity of the electro- 
 magnetic waves are the same. 
 
 Hertz constructed a large prism of pitch and with it found 
 that the electromagnetic waves are refracted like light waves. 
 The index of refraction which he determined was of the same 
 order of magnitude as that obtained for similar substances 
 with light. 
 
 The disturbances in the spark at which the electromagnetic 
 waves originate are all in one line, and so the waves which 
 come out from the origin are polarized. Hertz proved this to 
 be the case by interposing in the path of the wave a screen 
 made of a large number of parallel wires set at small dis- 
 tances from each other. When these wires were set parallel 
 with the direction of the spark they acted as a conducting 
 surface to absorb or reflect the vibrations. When they were set 
 perpendicularly to the vibrations, their effect was not nearly 
 so marked, and the waves passed through the screen. The 
 polarization of the electromagnetic waves, and the complete 
 analogy between their behavior and that of light waves, was 
 shown by the experiments of Trouton. Trouton allowed 
 electromagnetic waves to fall obliquely upon a thick stone 
 wall, and observed the reflected and refracted waves. He 
 found in general that the relative intensities of the reflected 
 and refracted waves depended upon the angle of incidence, and 
 also upon the angle between the direction of the electric vibra- 
 tion and the plane of incidence and that in particular, when the elec- 
 tric vibration was in the plane of incidence, and the an^le of inci- 
 dence had a certain value, corresponding to the polarizing angle 
 of light, the reflected wave entirely disappeared. In fact, the 
 behavior of these waves was exactly like that of polarized light 
 under the same conditions. 
 
 Numerous experiments have been made with electromag-- 
 netic waves to test Maxwell's relation between the dielectric
 
 THK ELECTRIC CURRENT. 887 
 
 constant of a substance and its index of refraction. The theory 
 has been fully confirmed. The reason why light wares do not 
 generally show an agreement with the theory seems to be that 
 the dielectric constant is not the same for the rapidly alter- 
 nating electromotive forces of the light waves as it is for the 
 slower alternations by which alone we can determine it. 
 
 By these and many similar experiments the properties of 
 the electromagnetic waves have been shown to agree precisely 
 with those deduced from the theory of Maxwell. We may 
 therefore consider this theory as confirmed, at least in its 
 essential features. We may consider it proved that electrical 
 action takes place in a universal medium, which is essentially 
 the same everywhere, except in so far as its properties are 
 modified in particular places by the material bodies there 
 present. From the general agreement in behavior between 
 electromagnetic waves and light waves, it may also be taken 
 as proved that this medium is the ether, with the properties 
 of which we have to some extent become acquainted in our 
 study of light. We may therefore adopt the electromagnetic 
 theory of light without reserve, and consider the subjects of 
 magnetism, electricity, and light, as forming parts of a gen- 
 eral science of the ether. 
 
 211. Pressure of Light. A remarkable conclusion of the 
 electromagnetic theory is that light will exert a pressure 
 against a surface upon which it falls. This conclusion was 
 unconfirmed for many years. An indirect proof of it was given 
 by Boltzmann, who showed that if it were correct, it could be 
 proved, by the methods of thermodynamics, that the rate of 
 radiation from a hot body should be proportional to the fourth 
 power of the absolute temperature. Now this law <>f radiation 
 had been announced by Stefan as the result of his examination 
 of the experimental facts connected with radiation; so that 
 this fact, at least, supported Maxwell's conclusion. Recently 
 .the Russian physicist Lebedew, and the American physicists 
 E. F. Nichols and Hull, have shown, by direct observation, 
 that light does exert a pressure such as the theory describes,
 
 338 THE ELECTRIC CURRENT. 
 
 and have measured its magnitude and found it to conform pre- 
 cisely to that predicted by the theory. 
 
 212. Magnetic Effect of Electric Convection, The concep- 
 tion that an electric current consists of a stream of electricity 
 along the conductor leads to the hypothesis that an electric 
 charge, when carried rapidly along, will set up a magnetic 
 field. This hypothesis was confirmed by the experiments of 
 Kowland, who observed a magnetic field in the neighborhood of 
 a charged disk kept in rapid rotation. Variants of Rowland's 
 experiment, executed by other observers, have yielded similar 
 results.
 
 KLKOTRIC IHSCHAKGE. 
 
 ELECTRIC DISCHARGE. 
 
 213. The Electric Spark. A critical examination of the 
 spark passing between oppositely charged bodies shows cer- 
 tain peculiarities in it, which of themselves would lead to 
 some modification of the theory of electricity which has 
 hitherto been presented. As the discharge by means of the 
 spark is variously modified, these peculiarities become more 
 and more prominent, and the necessity of some development cf 
 the theory of electricity becomes more and more apparent. 
 The most important advance that has been made in physical 
 science in recent years began with the study of these pecu- 
 liarities of the electric discharge. 
 
 The spark, as ordinarily seen, is a bright line or band cf 
 light, nearly straight when it is short, but broken or zig-zag 
 if its length is greater than two or three centimetres. By 
 observations of the spark in a rapidly revolving mirror. Fed- 
 dersen showed that, in ordinary circumstances, it consists of 
 a succession of sparks occurring at regular intervals, and 
 gradually diminishing in intensity. The whole spark, how- 
 ever, lasts for so short a time that, for many purposes, we 
 may consider it instantaneous. The special peculiarity of the 
 spark which was referred to in the preceding paragraph is that 
 the two ends of it, when carefully examined, do not seem to be 
 exactly alike. There is, in fact, a characteristic appearance at 
 the positive electrode which differs from that at the negative 
 electrode. This difference between the two ends of the spark 
 becomes more apparent when the discharge is modified, by 
 proper manipulation, into what is known as the brush dis- 
 charge. The brush discharge occurs between electrodes which 
 are at a considerable distance, perhaps 20 centimetres or so, 
 from each other. When it is established, a narrow band or 
 trunk of light originates at the positive electrode, and branches 
 out into innumerable fine lines of light, which cease to be visi- 
 ble before they reach the negative electrode. At the negative
 
 340 ELECTRIC DISCHARGE. 
 
 electrode there may generally be seen one or more short 
 brushes of light extending from the electrode for not more 
 than a centimetre. Between these and the positive brush there 
 is no visible evidence of the discharge at all. To ordinary 
 observation, at least, the brush discharge seems to be continu 
 ous. 
 
 By suitable manipulation the discharge may be made to 
 take still another form, called the glow discharge, in which 
 the characteristic difference between the positive and negative 
 ends of the discharge is also apparent. In this form of the 
 discharge, a faint luminous glow appears in a thin layer ovei 
 the positive electrode. Negative brushes, similar to those 
 already described, generally appear at the other electrode 
 These may sometimes be made to coalesce so as to form what 
 is called the negative glow. Even when this is done, how- 
 ever, the appearance of the negative glow is distinctly different 
 from that of the positive one. Except for these glows, no 
 light is produced by the discharge. The discharge in this case 
 also seems to be continuous. 
 
 In all our previoiis discussions we have treated the vitreoue 
 and resinous electricities, or the positive and negative electric 
 conditions, as if they were the precise counterparts of each 
 other, and differed in their properties only by being opposite tc 
 each other. In these various forms of discharge we have found 
 differences between the conditions at the two electrodes which 
 indicate that the differences between the two kinds of eleo 
 tricity may involve other features than those which have so 
 far been assumed. We shall find that this conclusion is con 
 firmed, and our knowledge of the peculiar characteristics oi 
 the two kinds of electricity very much enlarged, by the fur- 
 ther study of the discharge. 
 
 214. Discharge in Low Vacua. For the study of the dia 
 charge in rarefied gases we use what is called a vacuum tube, 
 that is, a glass tube or bulb from which the air or other gas oi 
 vapor which has filled it can be withdrawn by means of an ail 
 pump, and which is furnished with two terminals or electrodes 
 supported on platinum wires, generally sealed into the walls of
 
 ELECTRIC IJI*CHAR<JE. 341 
 
 the tube. In the future we shall call the electrode which is 
 joined to the positive pole of the machine, the anode, and that 
 joined to the negative pole of the machine, the cathode. When 
 the machine is in operation and the air is gradually with- 
 drawn from the tube., the discharge, which is at first an in- 
 termittent spark, becomes more and more frequent as the 
 racuum improves, until it is appreciably continuous. In this 
 3ondition almost the whole of the interior of the tube becomes 
 luminous with a faint rosy light. On examining the two 
 slectrodes we perceive very characteristic differences in the 
 discharge around them. It is not necessary to describe these 
 differences in detail. It is sufficient to say that the anode is 
 generally covered with a sheath or layer of rosy light appar- 
 ently similar to that in the body of the tube. This light, 
 which nearly fills the tube, is called the positive column. In 
 rery many cases it appears to be broken into saucer-shaped 
 layers or strata of light, separated by darker spaces. The 
 concave sides of these strata are turned toward the anode, 
 ind the light around the anode seems to be simply one of these 
 strata of modified form. The positive column terminates in 
 the tube, and is separated from the cathode by a dark space. 
 The cathode is surrounded by a bluish glow, extending to a 
 ?hort distance in all directions from it, and separated from it 
 by a non-luminous region. The color of the positive column, 
 and the size and distinctness of the strata, depend largely upon 
 the gas or vapor which is present in the tube. The relative 
 prominence of the different features of the discharge depends 
 upon the extent to which the exhaustion of the tube is carried. 
 As the exhaustion proceeds, the positive column diminishes in 
 length and brightness and the negative glow moves further 
 away from the cathode, thus increasing the width of the dark 
 space around it. 
 
 215. Discharge in High Vacua. In very high vacua, such 
 as can be obtained only by the use of the best air pumps, the 
 region occupied by the negative dark space is so enlarged as to 
 811 practically the whole of the tube. In this condition it 
 may be seen that this space is not entirely dark, but is filled
 
 <J42 ELECTRIC DISCHARGE. 
 
 with a faint bluish light. This light is indicative of the pres- 
 ance of the so-called cathode discharge. The cathode dis- 
 charge was discovered, and many of its properties were in- 
 vestigated, by Crookes, and the tubes used for its investigation 
 are frequently called Crookes' tubes. 
 
 Wherever the cathode discharge reaches the walls of the 
 tube, it sets up a peculiar phosphorescent illumination, the 
 3olor of which depends on the nature of the glass. In the soda 
 glass ordinarily employed, the color is a pale green. Similar 
 phosphorescence, with characteristic colors, is set up in very 
 many natural minerals, or chemical preparations. By the aid 
 jf a screen covered with a suitable preparation the path of the 
 uathode discharge may be studied. 
 
 By the direct examination of the light of the cathode dis- 
 charge, as well as by the use of phosphorescent screens, Crookes 
 showed that the discharge proceeds from all parts of the 
 cathode in lines normal to its surface. If, therefore, the 
 cathode is a flat plate which faces down the tube, a principal 
 part of the cathode discharge will be a nearly cylindrical 
 column lying around the axis of the tube. If the cathode is 
 a hollow spherical cup, with its concave side facing the tube, 
 the discharge will converge to the centre of the sphere, and 
 will diverge in a cone after passing through that centre. By 
 interposing an obstacle, like a sheet of metal, in the path of 
 the cathode discharge from a flat cathode, a sharply defined 
 shadow is cast on the phosphorescent screen placed beyond the 
 obstacle. 
 
 Crookes noticed that when the cathode discharge was di- 
 rected against light films of glass, they were moved as if a 
 blast of air had fallen on them. He constructed various me- 
 ehanisms, such as wheels furnished with vanes or paddles, and 
 found that, when the cathode discharge was directed against 
 them they were set in rotation as they would be by a blast 
 of air. He concluded that the cathode discharge contains mat- 
 ter moving with a high velocity. In his view this matter con- 
 sisted of the molecules of gas still remaining in the tubes,
 
 ELECTRIC DISCHARGE. 343 
 
 which were first negatively electrified by contact with the 
 cathode, and were then repelled from it. 
 
 Another experiment tried by Crookes was in harmony with 
 this view. In it a thin piece of platinum foil, placed at the 
 centre of a spherical cathode, was made red hot at the point 
 where the cathode discharge met it. The heat thus developed 
 can of course be explained by the impacts of the moving mat- 
 ter in the discharge. 
 
 When a magnet was brought near the tube, the cathode 
 discharge seemed to be affected by it. To test this more fully, 
 Crookes allowed the discharge from a flat cathode to pass 
 through a narrow slit in a sheet of metal, and then along a 
 phosphorescent screen, on which it developed a long narrow 
 band of light. This band of light was ordinarily straight, but 
 when a magnet was brought near the tube it was bent into a 
 curve. The direction of curvature depended on the direction 
 of the magnetic force which set it up, and was such as to in- 
 dicate that the stream of particles was negatively electrified. 
 
 This last conclusion is of great importance. It was sub- 
 sequently verified by an experiment of Perrin, which was re- 
 peated by J. J. Thomson. In this experiment the cathode dis- 
 charge was allowed to pass through a small opening in one 
 end of a brass box contained in the tube and joined with an 
 electroscope. As the discharge continued, the box became 
 more and more negatively electrified, as would be the case if it 
 were receiving continual accessions of negative charge in its 
 interior. 
 
 J. J. Thomson succeeded in showing that the discharge was 
 also acted on by an electric field as if it were a stream of nega- 
 tively charged particles. This cannot be done by a field set up 
 between electrodes which are outside the tube because the in- 
 duced charges in the walls of the tube destroy the field within 
 it; but by placing two flat electrodes within the tube, so that 
 the discharge could pass between them, and by using a very 
 highly exhausted tube, so that the residual gas in it was not 
 enough to rapidly discharge the electrodes, Thomson was able 
 to show that, when the electrodes were oppositely charged by
 
 344 ELECTRIC DISCHAKGK. 
 
 being joined to the poles of a battery, the cathode discharge 
 was deflected between them. The sense of the deflection indi- 
 cated that the discharge was repelled by the negative electrode. 
 
 It was shown by Hittorf that when a magnetic field which 
 is uniform throughout the whole length of the discharge is set 
 up around the tube, the cathode stream becomes the arc of a 
 circle. As was shown by Schuster, this observation is consis- 
 tent with, the hypothesis that the cathode stream is made up of 
 negatively charged particles moving with a high constant veloc- 
 ity. The amount of the deflection in a given magnetic field 
 depends upon the velocity of the particles, and upon the ratio 
 of their masses to the charges which they carry. By combining 
 the results of his own observations of the deflection in a mag- 
 netic field with those of his observations in an electric field, 
 J. J. Thomson was able to determine the velocity of the par- 
 ticles in the stream, and the ratio of their masses to their 
 charges. He found that the velocity was very great, in many 
 cases not less than one-third the velocity of light. The ratio 
 of the mass to the charge was about one thousandth part of 
 the ratio of the mass of the hydrogen atom to the ionic charge 
 which is associated with it in electrolysis. On the supposition 
 that the charge of each particle in the stream is equal to the 
 ionic charge, which seems from all the phenomena of electro- 
 lysis to be a natural unit of electricity, it would follow that 
 the mass of the particle is about the one thousandth part of 
 the mass of the hydrogen atom. This conclusion which, as 
 here stated, involves a certain element of hypothesis, was after- 
 wards confirmed by Thomson in several ways which will subse- 
 quently be described. 
 
 The effects produced by the cathode discharge were so strik- 
 ing that for a long time they occupied everyone's attention, so 
 that the anode discharge was entirely neglected. By a suitable 
 arrangement of the anode, however, it was found that streams 
 or discharges also proceed from it, which can be recognized by 
 their phosphorescent effects. The effects produced on these 
 anode discharges by the magnetic and electric fields were exam- 
 ined by W. Wien and by Ewers, and it was proved that the
 
 ELECTRIC DISCHARGE. 346 
 
 velocity of the particles in them is very much less than that 
 of the particles in the cathode discharge, and that the ratio of 
 the mass of these particles to their charges is of the same order 
 of magnitude as the ratio of the mass of an atom of an ordi- 
 nary metal to the ionic charge. These experiments proved at 
 the same time that the charges of these anode streams are 
 positive. It is surmised that these streams consist of posi- 
 tively charged atoms of metal torn off from the electrode. 
 
 216. Lcnard Rays. While experimenting with a cathode 
 discharge received upon an aluminium screen, Hertz discovered 
 that phosphorescent effects could be obtained behind the screen. 
 Guided by this discovery, Lenard constructed a vacuum tube, 
 at one end of which, opposite the flat cathode, the glass wall 
 was pierced by a small aperture or window, closed with a sheet 
 of aluminium foil. When the cathode discharge was directed 
 against this window, peculiar effects were obtained in the 
 region immediately around it, outside the tube. Among these 
 effects were the production of phosphorescence, and of photo- 
 graphic action on a photographic plate. In the experiment 
 with the photographic plate, the plate was screened from the 
 light of the tube by a covering of black paper, through which 
 the action penetrated. It was found that different bodies, 
 placed on the photographic plate, were penetrated by the action 
 in different degrees, it being in general true that the denser 
 bodies were less easily penetrated than those which were less 
 dense. The cause of these actions is evidently the cathode 
 discharge which has passed through the aluminium. 
 
 One of Lenard's most interesting experiments was made 
 with a tube of peculiar construction, which was divided into 
 two parts by a glass wall, furnished with an aluminium win- 
 dow. One of these tubes was furnished with anode and 
 cathode, and the air was exhausted from it to the particular 
 degree at which the cathode discharge is most strongly excited. 
 In the other tube, as perfect a vacuum was made as could be 
 made. With this arrangement, the cathode stream which 
 passed through the window proceeded in the second tube in a 
 perfect vacuum. Lenard examined the deflections of the cath-
 
 346 ELECTRIC DISCHARGE. 
 
 ode stream, in the second tube, produced by the magnetic and 
 electric fields, and obtained results for the velocity and for the 
 ratio of the mass to the charge which agree fairly well with 
 those of Thomson. 
 
 At the time Lenard discovered these effects he considered 
 the cathode discharge to be some sort of wave disturbance or 
 radiance in the ether, and spoke of the effects as due to rays. 
 These supposed rays were therefore called Lenard raya. On 
 the view of the nature of the cathode stream which we have 
 adopted, and to which Lenard finally came, they are not rays 
 in any proper sense. 
 
 217. Roentgen Rays. A series of additional effects pro- 
 duced by the discharge in a vacuum tube was discovered in 
 18!)6 by Roentgen. These effects are ascribed to what, aa will 
 be seen, we may properly call Roentgen rays. When the cath- 
 ode discharge falls on the wall of the tube, or better, when it 
 is concentrated by a spherical cathode upon a block of plati- 
 num, the place upon which it falls becomes the source of theae 
 rays. Some of them, at least, pass out through the walla of 
 the tube, and their effects may be studied outaide of it. The 
 most noticeable effect, and the one by which the rays were dis- 
 covered, is that of affecting the photographic plate, and of ex- 
 citing phosphorescence in many substances. The rays paas 
 through various substances which are opaque to light to a der 
 gree which seems to depend, other things being equal, upon the 
 densities of those substances. The photographic plate can 
 therefore be exposed to the rays while it is enclosed in a plate- 
 holder. Judging from the positions of the shadows cast on the 
 plate by various objects, Roentgen concluded that the action 
 travels from the point of origin in straight lines; and, from 
 the intensity of the photographic effect at different distances, 
 he concluded that the intensity of the action is inversely aa 
 the square of the distance from the point of origin. In theae 
 respects the action is comparable with light, and we therefore 
 designate as rays the lines along which it proceeds. In no 
 other respects have the Roentgen rays been proved to be the 
 same as light rays. There is no evidence of true reflection,
 
 ELECTRIC DISCHARGE. 347 
 
 though there is some diffuse reflection. Neither is there evi- 
 dence of refraction or of polarization. In a recent experiment 
 Haga and Wind have brought forward evidence which indi- 
 cates a diffraction, from which, if it is to be accepted, we may 
 conclude that the wave lengths in the rays are not more than 
 one five-thousandth as long as ordinary light waves. The hypo- 
 thesis that they are such waves is compatible with what is 
 known alxjut them. Even if we do not accept the evidence 
 offered to prove that they are diffracted, the wave theory of 
 their nature is still the most acceptable one. The waves are 
 probably set up by the sudden stoppage of the negative charges 
 in the cathode stream, and are therefore rather a series of 
 irregular pulses than a regular train of waves. 
 
 It -wa.5 shown by J. J. Thomson that when the Roentgen 
 rays pass through a mass of air they render it a conductor. 
 According to Thomson the conducting power is given to the 
 gas by an ionization of its atoms, that is, by a separation of 
 some of the atoms into positive and negative portions. 
 
 218. The Negative Electron. J. J. Thomson undertook an 
 investigation of the properties of ionized air, in order to deter- 
 mine the magnitude of the charges carried by the ions. Ruth- 
 erford had determined, in 1897, the velocity of the ions in an 
 electric field of definite strength. By ionizing air between two 
 electrodes, using the Roentgen rays for that purpose, and by 
 measuring the current transmitted by them between the elec- 
 trodes when their difference of potential was known, a quan- 
 tity was obtained equal to the number of the ions, multiplied 
 by the charge on each, multiplied by their velocity. The veloc- 
 ity being known from Rutherford's experiment, the charge 
 could be determined if the number of the ions could be deter- 
 mined. To accomplish this Thomson used a discovery of C. T. 
 R. Wilson that when dust-free air, saturated with water vapor, 
 is ionized, and then suddenly expanded to a certain degree, leas 
 than that required to produce condensation in ordinary air, 
 condensation of water occurs on the negative ions. By observ- 
 ing the rate at which the fog thus formed settled, the size of 
 the drops was calculated, and from an observation of the quan-
 
 348 ELECTRIC DISCHARGE. 
 
 tity of water condensed, the number of drops and so of the 
 negative ions was determined. From the data thus obtained 
 it was shown that the charge of the negative ion is the same 
 as that belonging to the hydrogen atom, and called the ionic 
 charge. 
 
 It was shown by Elster and Geitel that when ultra-violet 
 light falls en a charged conductor in air, the conductor is 
 speedily discharged if its charge is negative, while it is not 
 discharged if its charge is positive. The effect seems to be to 
 excite conditions in which negative charge is liberated from 
 the body, for if the light falls on an uncharged conductor, it 
 will gradually become positively charged. This action is best 
 obtained with clean zinc. 
 
 It was shown by the same observers that the discharge 
 from a plate of zinc is diminished, and finally checked alto- 
 gether, if the zinc is in a sufficiently strong magnetic field, the 
 lines of force of which are parallel with its surface. The 
 experiment succeeds only when the zinc is in a vacuum. On 
 the supposition that the escape of the negative charge takes 
 place by the emission of negative ions from the surface, 
 Thomson showed that the effect of the magnetic field ought to 
 be to change the straight paths of the ions into cycloids, and 
 that if the discharge is taking place between the zinc and an 
 electrode placed parallel with it and near it, a current ought 
 to pass between the plates, of which the intensity will remain 
 appreciably the same so long as the cycloidal paths of the 
 ions meet the second plate, but will change abruptly, or at 
 least very rapidly, for such a strength of the magnetic field 
 that most of the cycloidal paths do not meet the second plate. 
 By trial Thomson convinced himself that such an abrupt 
 change in the current occurred, and that his hypothesis was 
 therefore justified. From the distance between the plates, he 
 determined the magnitude of the cycloidal paths, and from 
 this and the observed values of the difference of potential 
 between the plates, and the strength of the magnetic field, he 
 computed the ratio of the mass of the ion to the charge upon 
 it. He found this ratio to be the same as that which he had
 
 BLKCTRIC DISCHARGE. 349 
 
 determined for the particles in the cathode stream. By con- 
 densing water drops on the ions, in the way already explained, 
 he determined their number, and hence determined the charge 
 on the ion, which he found in this case also to be equal to the 
 ionic charge of the hydrogen atom. In this way, without hy- 
 pothesis, he showed that the mass of the negative ion which 
 is associated with an ionic charge is about the one-thousandth 
 part of the mass of an hydrogen atom. It can hardly be 
 doubted that the negative charges in the cathode stream are 
 also ionic charges-, and that the mass of the particle is far 
 less than that of the hydrogen atom. 
 
 The negative ions thus recognized must be distinguished 
 from the ions in electrolysis, which are masses of matter con- 
 sisting of one or more atoms, and associated with a number 
 of ionic charges equal to the valency. To denote this differ- 
 ence, we may call the negative ions of the cathode stream, or 
 those produced in a gas by the Roentgen rays, or otherwise, 
 by the name electrons. 
 
 To describe these results, Thomson supposes that the atoms 
 of a gas consist of a positive portion joined to a number of 
 negative electrons, which neutralize the positive charge, and 
 that, when the gas is ionized, some of its atoms liberate each 
 one electron, and become themselves positively charged. 
 
 210. Becquerel Rays. Not long after the discovery of the 
 Roentgen rays the French physicist Becquerel discovered that 
 similar effects to those produced by them could be obtained 
 from masses of uranium or of ores containing uranium. The 
 activity of the uranium was very slight, and several days 
 were required to obtain the photographic effects by which the 
 properties of these rays were studied. They were found to be 
 in general similar to those of the Roentgen rays, but certain 
 differences were noticed, which could not be fully studied on 
 account of the feebleness of the action. The action, however, 
 was considered to be due to rays like the Roentgen rays. 
 
 220. Radium and Radioactive Substances. M. and Mad- 
 ame Curie found that certain uranium-bearing minerals were 
 more active than metallic uranium. As the result of long
 
 350 ELECTRIC DISCHARGE. 
 
 continued labor, they succeeded in isolating from pitch- 
 blende, an ore of uranium, a small quantity of a salt of a new 
 element, which they named radium. 
 
 Thiselement exhibits properties similar to those of uranium, 
 but to a far greater degree. They are called generally radio- 
 active properties. The most delicate and ready way of detecting 
 radioactivity is by the use of a sensitive electroscope, which 
 will gradually be discharged if the air around it is ionized by 
 the radioactive body. Thorium, among the elements already 
 known, was found also to be radioactive, and the discovery 
 of several other new radioactive elements has been announced. 
 It is not yet determined whether these other elements have a 
 real existence, or whether the effects attributed to them are not 
 rather due to radium or to a product of its disintegration. 
 
 The properties of thorium and radium have been very care- 
 fully studied by Rutherford and Soddy and their associates. 
 The conclusions which they have drawn are too complicated 
 for a brief description, but certain of their results and a gen- 
 eral statement about the others can be given. They conclude 
 that the atoms of these bodies are one by one disintegrating. 
 This disintegration is accomplished by sending off negative 
 electrons, and also positively charged bodies. Only a rela- 
 tively very small number of atoms break up at once, though 
 the actual number is very large; sufficiently large for their 
 emission to constitute streams of negative electrons and posi- 
 tive bodies. The stream of negative electrons can be detected 
 by their behavior in a magnetic field. It is exactly like the 
 negative stream from the cathode. The stream of positively 
 charged bodies is also affected by the magnetic field, in such a 
 way as to show that the velocities of those bodies are much 
 less than the velocities of the negative electrons, and that their 
 masses are much greater. This conclusion follows also from 
 the fact that the positive streams are intercepted almost en- 
 tirely by a very slight obstacle, like a sheet of paper, while 
 the negative streams have much greater penetrating power. A 
 third set of streams or rays has been recognized which can 
 penetrate through several inches of iron or lead. In the case
 
 ELECTRIC DISCHARGE. 361 
 
 of radium another set of bodies, which Rutherford calls 
 the emanation, leaves the mass. This emanation seems to be 
 the remnants of the radium atoms, after they have undergone 
 the first disintegration. The emanation undergoes another 
 disintegration, during which negative electrons and positively 
 charged bodies are again emitted. The experiments of Ram- 
 say and Soddy indicate that one of the products of these dis- 
 integrations, probably the positively charged bodies first 
 emitted, are atoms of the element helium. 
 
 Rutherford and Soddy proved with thorium that it was 
 possible to separate it by chemical processes into two parts, 
 one of which emitted negative electrons and positively charged 
 bodies, while the other emitted only the latter. With lapse of 
 time both portions acquired the normal type of radioactivity. 
 We can explain these results only by supposing that a contin- 
 ual alteration is going on in the thorium of one class of atoms 
 into another, and that the atoms thus formed are continually 
 breaking down into the products which indicate radioactivity. 
 The normal radioactivity of the thorium is that which is 
 reached when as many active atoms are being produced as are 
 being destroyed. Not only in the case of thorium, but in the 
 case of the other radioactive elements, especially of radium, 
 observation indicates a gradual disintegration and destruction 
 of these elements. Rutherford calculates that the life of a 
 mass of radium is about two thousand years. Tha* of thorium, 
 which disintegrates much less rapidly, is many millions of 
 years. 
 
 Radioactive properties, which have been shown to be simi- 
 lar to those of the emanation of radium, are exhibited by 
 specimens of water drawn from wells in all parts of the world. 
 We may conclude from this observation that radioactive bodies 
 are very widely distributed in the earth's crust. 
 
 221. Electric Theories of Matter. Kaufmann has exam- 
 ined by experiment the effect produced by a magnetic field on 
 those rays from radium which correspond to the cathode 
 stream. He finds that under the action of the field the stream 
 spreads out into a gradually widening band. By studying the
 
 352 ELECTRIC DI8CHARQK. 
 
 curvatures of different parts of the stream, he finds that they 
 are consistent with the supposition that the electrons in the 
 stream have no mass. It was shown by J. J. Thomson from 
 Maxwell's theory of electricity that when a charged sphere is 
 moving rapidly through the ether, and is acted on by a force 
 tending to change its motion, it will behave as if its masa 
 were greater than the mass of the sphere, by an amount which 
 depends on the electric charge upon it, and the velocity with 
 which it is moving. This general conclusion, that a moving 
 charge will have an apparent inertia, due to its interaction 
 with the ether, has been confirmed by other investigators, and 
 especially by Abraham, with whose equations Kaufmann inter- 
 preted his results. Kaufmann finds that the paths of the 
 electrons sent out by the radium are such as to indicate that 
 the mass of the electron depends upon its velocity, and in such 
 a way as it would if the electron possessed no mass except 
 the apparent mass due to its electric charge. If this result is 
 accepted, we must conclude that the electron is a small por- 
 tion of electricity, separate from matter. 
 
 The apparent mass which a moving electric charge can have 
 indicates a theory of matter in which the material atoms are 
 assumed to be assemblages of electrons, moving around each 
 other in very small orbits, but with enormous velocity. The 
 ether is assumed to be a continuous medium, possessing a 
 peculiar elastic reaction to torques or twists. Theories rest- 
 ing on this general supposition have been developed by H. A. 
 Lorentz and by Larmor, and have been shown capable of ac- 
 counting for most, if not all of the phenomena of light and 
 electricity, in their relations to moving, as well as to station- 
 ary bodies. 
 
 On this general theory a radioactive body is one in which 
 the electrons in the atom are so numerous and are moving in 
 such manner that the bonds which unite them together are not 
 always stable. This being so, if the limit of stability is passed, 
 Uhe atom may break up, one of the results of its breaking up 
 being generally the emission of at least one electron.
 
 PLAT 
 
 Rig. 2

 
 i
 
 PLATE 5 
 
 Fig. 42 
 
 Fig. 43 
 
 Fig. 44 
 
 Fig- 45
 
 PLATE 6 
 
 Fig. 46 
 
 F '&- 47
 
 PL 
 
 ATE 7 
 
 Fig- 55 
 
 Fig. 56 
 
 Fig- 59 
 
 Fig. 60
 
 PLATE 8 
 
 Fig. 62 
 
 Fi K . 68
 
 Fig. 69
 
 
 This book is DUE on the last date stamped bel( 
 
 I 
 
 NOV 2 8 1994 
 OCT i'i 
 
 QCT 23 1940 
 
 A *UG 7 7 1942 
 DEC 2 1 1^42 
 
 NOV l 4 1948 
 MAR 2 8-195Q 
 
 Form L-9-15wi-7,'32
 
 QC31 
 
 M27s Magie - 
 
 Syllabus__of_ 
 a course of 
 lectures on 
 
 physicsT" 
 
 Q.CI3 
 
 UMIVERSITY of CALIFORNIA 
 
 < TBRABY