s PC; o g> 00 PQ fc O H I ( OQ w GO If i Bopt. Mocli, Bng. Dopt. Mooh. ESng. THE Dopt.Moobi.Bng. GALVANOMETER, AND ITS USES. A MANUAL FOR ELECTRICIANS AND STUDENTS. BY C. H. HASKINS. THIRD EDITION. WITH ILLUSTRATIONS. NEW YORK: D. VAN NOSTRAND, PUBLISHER, 23 MURRAY ST. AND 27 WARREN ST. 1884. Entered according to Act c' Congress, in the year 1881, by D. VAN NOSTRAND, In the Office of the Librarian of Congress, at Washington. Electrotyped by SMITH & McDoucAL, 82 Beekman St. Dopt. Mooh. PREFACE. r ~PHE author has endeavored to present within the limits of this little volume, in a condensed form, a statement of electrical laws and rules for measurement, written in a plain, brief manner, to- gether with such tables and simple formulae as will enable any student, with a little practice, to make land line measurements with dispatch and accu- racy. He trusts, also, that it will be valuable to his more experienced brethren as a book of reference, obviating the necessity of search through the works of different authors for what will here be found in a condensed and convenient form. Considerable matter relative to construction and maintenance of lines, care of battery, etc., which the author had prepared, has been omitted in order that the book might be confined to the original plan that of a manual of measurement. The author has no apologies to make for any errors of omission or commission. If this effort should iv PREFACE. stimulate another to the preparation of a more com- plete work, the profession will be benefited thereby, and the author fully compensated for his labors. To the gentlemen who have furnished formulae, and especially to his friend F. L. POPE, the author returns his sincere thanks. CONTENTS. EXPLANATORY CHAPTER. PAGE Resistance of Conductors 7 Units of Measurement 8 Electromotive Force 8 Quantity 9 Cause of the Flow of the Current 9 Law Regulating the Flow of the Current 10 Quantity and Tension of Batteries 10 Proportioning Battery to Line 12 The Speed of the Current 15 Effect of Atmospheric Moisture on Insulation 17 Joint Resistance of Lines 18 Galvanometers 19 Rheostats 20 Shunts 20 CHAPTER II. Tangent Galvanometer 21 Connections for Measurement 23 Testing for Insulation 24 Gaugain Galvanometer 25 Differential Galvanometer 26 Wheatstone Bridge 27 Sine Galvanometer 32 vi CONTENTS. CHAPTER III. Formulae to measure Electromotive Force and Internal ?A Resistance of a Battery By method of Equal Deflections.. . By Direct Measurement Internal Resistance of Battery Mance's Formulae 3 . Measurement of Electromotive Force by Comparison 36 Measurement of Insulation 3 6 Testing for Conductivity 38 Line Resistance measured from an Intermediate Sta- tion 38 Corrected Line Resistance, by Schwendler 39 Testing for Grounds 40 By Loop Test 4I Testing for Crosses 42 Testing Bad Grounds 45 Shunts 45 Units of Measurement 6 Weight of Insulated Office Wire 47 Specific Resistance of Different Metals 47 Table of Weight and Resistance of Copper Wires 48 Electromotive Force of Batteries 49 Table of Natural Tangents 50 English and French Measure 53 Table of Sines and Tangents 54 Table of the Sizes, Weights, Resistances, and Breaking Strain of Iron Wires: 56 Table of Squares, Cubes, Square and Cube Roots 57 APPENDIX 77 J EXPLANATORY CHAPTER. To enable the student to work understanding^ in salvanometric measurements, it is necessary that Should comprehend the laws upon which such urements are based. These laws we will state, in as few and simple words as possible ; and the student, if he masters them thor- oughly, will be enabled to understand readily what ^Resistance of Conductors. -All conductors offer a certain degree of resistance to the passage o the electric current through their substance resistance varies with different materials. The follow- ing table illustrates the comparative resistances of the most common metals to the passage of the current, ;ilver being taken as the standard. The measure ments are taken at 32 Fahrenheit. Silver, hard 100 Copper, hard 100 Gold, hard 128 Iron 594 Tin 8 9 Lead i Brass 45 German Silver '1282 Platinum and silver.. 1493 Mercury 5 8l 5 Jenkin. The resistance of metals increases with their tner mometric heat. For instance, iron wire increases abou 35 per cent, for each degree Fahrenheit above 60 . The conductivity (which is the reverse of resistance of soft iron wire, suitable for telegraph purposes, i about \ that of commercial copper wire. The conductivity of any wire increases as its weigh per foot. . The resistance of any iron wire, per mile, is loun by dividing 360,000 by the square of its diameter, Th 8 MEASUREMENT FORCE. quotient will be in ohms at 60 Fahrenheit. This is ^60 ooo expressed thus ' , d being the diameter of the wire. Or it may be measured, as hereafter explained. Knowing, then, the resistance, per mile, that a given wire offers, or should offer, to the passage of a current, we have a basis for comparison, and can readily ascer- tain its electrical condition, and, when defective, apply a remedy where and when required. 2. Units of Measurement. To determine accu- rately the amount or degree of resistance, we must have some standard that will serve us as the inch does in the measurement of distance, or a grain in estimating weight. Several of these standards have been made. (See Table of Units of Resistance^) But two standards, however, are now in general use, and these two only shall we note here. The Ohm is the standard unit, adopted by the British Association, and in general use in this country. It is equal to the resistance of a prism of pure mercury of i square millimetre section and 1.0486 metres length, at 32 F. The Siemens Unit is equal to the resistance of a prism of pure mercury of i square millimetre section, and i metre long, at 32 F. The Siemens Unit is in general use on the European continent, and to a con- siderable degree in the United States. To convert Siemens into Ohms, multiply by the decimal .9536. To convert Ohms into Siemens, multiply by 1.0486. Having found our units of measurement, we will now consider the laws governing the flow of the current.* 3. Electromotive Force. The power which a cell of battery possesses of causing the transfer of its cur- * In examples given in this work, the term u unit" is used as a "unit of measurement," and may be read either as an Ohm or Siemens' unit. When absolute measurements are given, the word Ohm or Siemens is always given to distinguish the kind of unit meant. QUANTITY THE FLOW OF THE CURRENT. rent from one place to another is its electromotive force. In other words, the electromotive force of a current is its power of overcoming resistance its energy. To use a familiar comparison, electromotive force is to a current what pressure is to steam. 4. Quantity. The amount of current that is evolved in a given time by a cell of battery is termed its quan- tity. This amount may be increased or diminished by lessening or increasing the resistance in circuit both in the battery and external thereto. 5. Cause of the Flow of the Current. While chemical action is going on in a battery, the current is being constantly accumulated, or piled up, at the posi- tive pole of the battery, with a vigor depending en- tirely upon the electromotive force of the battery. The old theory, that the current thus accumulated flowed along the line wire to the ground at the dis- tant end, and back to the other terminal of the battery through the ground, is now generally abandoned, and the theory of M. Gavarret accepted as correct. He teaches that the earth being a common recep- tacle and reservoir of electricity hence termed the "common reservoir" has no electrical tension. When the current in a battery accumulates at the positive pole of the battery, its tension is greater than that of the earth. Now connect a wire from each end of the battery to the ground. If these wires are short and thick, thus having practically no resistance, the current will flow from the positive pole of the battery through the wire to the earth with a degree of energy depending upon its excess of tension over that of the earth. This action will continue, the earth receiving and absorbing from the positive pole of the battery where the tension is greater, and giving to the nega- tive pole where the tension is less, until the action of the battery ceases from exhaustion, and the tension of the battery and the earth are equal. The effort, then, of the current evolved by the bat- io FLOW OF CURRENT QUANTITY OF BATTERIES. tery, is to equalize the tension or pressure, and produce an electrical equilibrium. This same inequality of ten- sion between the clouds and the earth, is the cause of lightning-storms. 6. L,aw regulating the Flow of the Current. If, in the case above supposed, the wires connecting i the battery with the earth, were so long, or so thin, as to interpose considerable resistance to the passage of the current, the flow would be less in amount, and the chemical action of the battery less rapid the current being dammed, or choked back, by the resistance to its passage, The amount of current, then, that will flow to line, depends upon and is proportionate to the resistance of the circuit, Hence Ohms law, which is the founda- tion of all electrical measurements. Ohms law is stated thus : C being the amount of current, E the electromotive force, and R the resistance in the circuit. or, the current flowing to line, equals the electromotive force divided by the resistance. 7. Quantity and Tension of Batteries. The amount of current generated by a cell of battery de- pends upon the size of its plates the tension, upon the number of cells. Hence, a battery of forty cells, while giving forty times the electromotive force, or tension* of one cell, will furnish only the same amount or quantity of current. Each cup of battery has its own quantity, which is urged forward to the next cell, by virtue of its own * Electro-motive force may be defined as tension in a state of motion; and tension, as electromotive force in a state virest, QUANTITY AND TENSION OF BATTERIES. 11 electromotive force. The current from the cell at the positive pole of the battery, is pushed out to the line- wire, the current from the next cell taking its place the last cell being supplied, through the ground-wire, from the earth. But the current from each cell, when carried forward to its neighbor, parts with its energy or electromotive force ; and the current from the first cell is pushed forward to line, and thence to the ground at the further end of the line, with all the electromo- tive force, or energy, of all the cells. The amount of current flowing to line, from a given battery, is regulated, as we have seen, by the resistance of the circuit and in this must be included the inter- nal resistance of each cell of the battery. But, with a given total resistance, the flow of current may be in- creased : i st. By increasing the size of the cells, and thus de- creasing the total resistance in the circuit ; and, 2d. By adding more cells. If we consider only the internal resistance of the cells, the addition of more cells will not increase the current. Thus, if we have i cell of battery, with an internal resistance of 2 units, and an electromotive force of, say, 10, and the poles connected by a wire having practically no resistance, the proportion of current flowing to line will be 5 . Now take 5 cells, E 5 x 10 = 50 _ ~R 5 x 2 = 10 ~~ > * The product is still 5, because the divisor and divi- dend are both changed in the same ratio. 12 PROPORTIONING BATTERY TO LINE. But suppose the resistance of the line-wire to be 5 units, represented by r. Then with i cell we have, , = IO -=='43. and with 5 cells, or nearly three times as much current as before. 8. Proportioning Battery to Line. It will be seen, therefore, that the proper method for adapting a battery to any line, with any instruments, is to select a form of battery giving sufficient current to saturate the magnets of the relays to their maximum, and then add a sufficient number of cells, to drive it with the necessary force to produce the desired effect. The sum of the resistances of all the instruments in a circuit, should, as nearly as practicable, balance the resistance of the line-wire. In no case should they ex- ceed the wire resistance. " IV hen the resistance of the coils of the electro-magnet is equal to the resistance of the rest of the circuit, i. e., the conducting wire and battery, the magnetic force is a max- imum" (NOAD's Text Book.) "The application of this law to a telegraphic circuit would be to make the sum of the resistances of all the magnet-coils in circuit, equal to the resistance of the line and batteries ; but as in practice the resistance of a telegraphic circuit varies, being considerably reduced by defective insulation, the total resistance of the in- struments should be less than that of the line when in good condition, to attain the best results during unfa- vorable weather." (POPE'S Modern Practice, page 152.) A safe, practical rule for determining the number of cups of main battery needed, is to call the resistance of the relays equal to the line-wire, and use one cup of Grove, or bichromate of potash (electropoion), or two PROPORTIONING BATTERY TO LINE. 13 cells of Calland or Hill battery to each 100 units re- sistance in the circuit. Thus, for example : suppose a circuit of 200 miles No. 9 wire. This will measure 20 units to the mile, or . .4000 units. 8 relays in circuit, 500 units each 4000 " Total resistance 8000 " For Grove or bichromate, divide by 100, and we get 80 cups battery, 40 at each terminal of the wire. For sulphate copper battery, divide by 50, and we find that we need 160 cups of battery, 80 at each end. These figures are true only of single wires. When several wires are worked from one battery, the total amount of battery used is of course much less. The number of lines of about equal resistance that may be worked from the same battery varies greatly with different batteries, depending upon the chemical en- ergy of the battery, the quantity of current evolved by it, and its liability to polarization. Thus, the bichromate batteries of 80 cups, that we determined upon for our 8000 unit line, will furnish current enough for several lines of the same resistance. But as we add wires we lessen the external resistance, until, owing to the slight resistance, the action is so rapid that the battery polarizes and action ceases. Practically, not over six to eight lines can be worked from this battery. With Grove it is different. Lines may be added to a Grove battery, with perfect safety, up to forty or fifty. The sulphate copper batteries have too much inter- nal resistance to allow the addition of many wires. Three lines are worked from sulphate copper batteries having jars six inches in diameter. In all cases, the expenditure or consumption of ma- terial in a battery is in exact proportion to the work PROPORTIONING BATTERY TO LINE. done ; that is, to the current evolved. Therefore, the expenditure of battery material, with three lines to a battery, is three times that of one line to the same battery, and there is no economy in working several lines from one battery, except in room occupied. The disadvantage of working several lines from one battery is the interference of the circuits with each other in bad weather. Sometimes this interference is quite serious. A greater number of cells of battery than is absolutely needed to work the circuit promptly, is in- jurious, and in wet weather it tends to make the line work badly. Indeed, the effect of an excess of battery is often as detrimental to a line, in wet weather, as bad insulation, or a conducting wire of too high resistance. To explain this, the reader will remember that the difference in resistance between the line-wire and the insulators, is the margin upon which the line is worked. If that margin is diminished by bad insulation, a por- tion of the current escapes to the ground. If the re- sistance of the line-wire is increased, the margin is again diminished, and a portion of the current escapes as before. Now, the higher the tension of the battery-current, the more readily will it escape from the line, because the greater is its power to overcome resistance. The effect, then, upon a wet day, is the same, upon the line, whether the insulation is defective, or the battery too large.* Thus it will be seen that adding more battery to a heavy-working line, will often, in wet weather, make it work worse than before. And it will also be noted, that relays of high resistance, as they add to the resist- ance of the line-wire, decrease the working margin * If, however, the escape is not very serious, it is sometimes beneficial to add battery to sending end until you have so high a current that, in addition to the large amount escaping, enough will reach the further end to work the relay. In this case the receiver must not attempt to 11 break." J THE SPEED OF THE CURRENT. between the line and insulator resistance, and thus increase the tendency of the current to escape. When the relays of a line, of say 200 miles, are pro- portioned to the line itself, that is, do not exceed the line-wire resistance, and the battery is adapted to the circuit by the rule above given, if the line is well insu- lated, it will work well, in very bad weather, when a similar line, equipped with high resistance relays, or an excess of battery, will scarcely work at all.* 9. The Speed of the Current. Many experi- ments have been made to determine the speed of trans- mission of the electric current or influence through a wire. The first measurement, made by Wheatstone in 1833, showing a speed of 288,000 miles per second, was taken as the speed of the current under all circum- stances. Subsequent measurements, however, showed such variable results, that confidence in Wheatstone's figures was much shaken. The following table shows the rates of speed noted by various experimenters : Date. Length of Circuit. Velocity Per Second. Observer. Remarks. 1833 .5 mile 288,000 Wheatstone Leyden Jar. 1849 880 18,700 Walker Relay. k 59> 16,000 u u 4 607 28,500 Mitchell u 1850 260 iron 60,000 Fizeau & Gounelle Galvanometer needle. 1 130 copper 114,000 41 U U U ' 1045 15,000 Gould Relay. 1 447 17,000 Walker " 1854 230 104 iron 13,000 115,000 Guilleman & Burnouf Chemical Telegraph. Galvanometer needle. * A line that is suffering from bad escape may be worked much better for through business in bad weather, by switching off the battery at the receiving end of the line, and receiving only by the sending office cur- rent. Because, if there are heavy escapes on the line, the sending opera- tor does not break all the current from the battery at the receiving end of the line, thus leaving the receiving relay partially magnetized constantly, and consequently less sensitive to the small portion of current from the sender that reaches the receiving station. Whereas, if the only current on the wire is that of the sender, the portion of his current that does reach the other end of the line comes clear and sharp. 16 THE SPEED OF THE CURRENT. The above table gives no data from which to make any calculation or deduce any law. It only serves to show how varied were the results obtained. To be of any benefit, the experimenters should have given the conductivity, per mile, of the wire used, the form of battery, and the number of cells used in each case. The English, French, and German experimenters use, also, miles of different lengths. Thus, for land lines, we were without any reliable data until 1869, when Prof. G. W. Hough, of Dudley Observatory, and Mr. C. S. Jones, of Albany, New York, undertook a series of experiments to determine the velocity of the current. Mr. Jones, manager of the Western Union office, by looping wires to Boston, De- troit, New York, and Buffalo, obtained circuits of suffi- cient length to test the question thoroughly. As Grove battery was used, and the line-wire was galvanized No. 9, we have some data from which to draw satisfactory conclusions. On the 27th of May, Messrs. Hough and Jones ob- tained the following results : Velocity per Second. No. Cells of Battery. Length of Circuit. 10,200 miles. 20,000 u 29,450 " 18,200 r 160 295 295 400 miles. 400 l 400 l IOOO Prof. Hough remarks, in relation to the above re- sults : " An inspection of these results show at a glance, that the velocity increases with the number of battery elements employed ; also, for the same battery it de- creases with the length of the circuit." From these figures we deduce these facts : First. That with a given resistance, the speed in- creases with the increase of the electromotive force, EFFECT OF MOISTURE ON INSULATION. 17 i. e., the power to overcome resistance and urge the current forward. Second. That with a given battery, the speed of the current varies as the resistance to be overcome in the circuit varies. And these facts explain why Wheatstone obtained a velocity of 288,000 miles per second, when Hough and Jones show less than 30,000 miles as their greatest ve- locity. Wheatstone used a Leyden jar, discharging a current therefrom of the highest tension, and having but a half mile of resistance of circuit, while the Albany experiments were made through long circuits, with battery-currents of comparatively very low tension. IO. Effect of At mosplieric Moisture on Insula- tion. The effect of a moist or dry atmosphere on line measurements is too generally overlooked, especially on lines insulated with glass. As an insulator, glass is probably the best in dry weather, as it is certainly one of the worst when surrounded by a saturated atmos- phere. Being- an excellent radiator of heat, the glass insulator readily gives it out to the passing breeze, and speedily becomes cold, when the particles of moisture held in suspension in the air are deposited in a con- densed form on the surface of the glass, thus forming a film of water, through which the current escapes. Again, even with the most perfect insulation, with a saturated atmosphere, a line will show considerable general escape. Frequently, a lower degree of insula- tion is shown just before a heavy rain than after it be- gins falling. And although the most careful experi- ments fail to show any escape through the atmosphere from one particle of moisture to another, yet this heavy escape is often noted, as above stated. When the rain begins falling, the moisture being condensed into drops, the continuity of the moisture seems destroyed, and the escape consequently checked. Whenever practicable, then, the percentage of sat- i8 JOINT RESISTANCE OF LINES. uration of the atmosphere, shown by a hygrometer, should be recorded, with the measurement, for future use, in comparing the condition of a line at different times. 11. Joint Resistance of Lines. By attaching a second wire to a battery, a second path is provided for the current, and the resistance to its flow therefore lessened. If both wires are of exactly the same resist- ance, the effect is the same as if the first wire were taken off and replaced with one of just double the ca- pacity, or weight per foot, or one of half its length of the same capacity. The joint resistance that is, the resistance of all the wires attached to a battery to the flow of its current, may be found as follows : If the resistance of the wires is equal, the sum of the resistances, divided by the square of their number, or the resistance of one wire, divided by the number of wires, will give the joint resistance. Thus, four wires of 100 ohms each : ist method, 2d method, 4?? = 25 ohms. 16 I?? = 25 ohms. 4 When the resistance of two wires is unequal, the resistance may be found by dividing their product by their sum. Suppose two wires of 200 and 300 ohms resistance, 200 x 300 = 60,000 =I2QQhms> 200 -f 300 = 500 When more than two wires, find the joint resistance of the first two, as above ; then take the result, consid- ering it as a single wire, and proceed in the same manner. GALVANOMETERS. 19 Suppose we have three wires, of 200, 300, and 280 ohms. The joint resistance of the first and second is, as we have seen, 120 ohms. 120 x 280 33,600 , = 84 ohms, 1 20 + 280 400 the joint resistance of the three wires. Or, it may be obtained by the following formula : i st. Calling the three wires a, b, c, abc ab -f ac + be ' 200 x 300 x 280 = 84 ohms. 200 x 300 + 200 x 280 + 300 x 280 2d. Divide one by the sum of the reciprocals* of their several resistances. i The reciprocal of 200 is ................. 005 " 300 is ................. 00333 + " " " 280 is ................. 00357 + .01190 and, = 84 ohms. .01190 12. Galvanometers. A magnetic needle will be deflected from the north by a current of electricity flowing in a conductor placed parallel to it. If the conductor is placed on top of the needle, and parallel, when the positive pole of the battery is attached to the south end of the conductor, and its negative end to the * The reciprocal of a number is the quotient of i divided by that number. Thus the reciprocal of 2, is ; of 20, is -^- , 20 RHEOSTATS SHUNTS. north end, the current will flow northward, and the north end of the needle will be deflected to the left. Reverse the current, and the north end will be deflect- ed to the right. If the conductor is placed under the needle, the action will be in'the reverse direction. The degree of deflection will be proportional to the strength of the current. If the conductor is carried over the face of the needle, and back underneath, the effect will be doubled. Wind the wire in several convolutions parallel to the needle, making a helix, with the needle in the centre, and you have the strongest form of galvanometer. The finer the wire, and the greater the number of convolutions, the more sensitive will the needle be to the current passing through the wire of the helix. 13. Rheostats. By using fine German-silver or silver and platinum wire, the resistance of which is not affected by changes of temperature, we may make a series of coils whose resistance is equivalent to that of any number of miles, or fractions of miles. A num- ber of these coils, arranged to be thrown in and out of circuit at pleasure, is termed a Rheostat. 14. Shunts. It is sometimes necessary, in meas- urements, to divert a portion of the current from a galvanometer, to reduce the deflection of the needle within proper limits. In this case, a wire or coil of greater or less resistance is used, which is connected to both posts of the instrument to be shunted. The current will then divide inversely as the resistance of the two paths or routes open to it. Suppose a galvan- ometer coil to have a resistance of 99 ohms, and the shunt-wire of i ohm. Then -ffo of the current would pass through the shunt wire, and T fg- through the galvanometer. In this case, the result shown by the galvanometer must be multiplied by the shunt to obtain the correct answer. See rules for making shunts. TANGENT GALVANOMETER. 21 CHAPTER II. 1. Tangent Galvanometer. "The intensity of the current, as measured by the tangent galvanometer, is proportioned to the tangent of angle of deflection " of the needle. " Bradley 's Improved Tangent Galvanometer," is the only one in use in the United States for line work, and is undoubtedly the most accurate tangent instru- ment made. We quote the description from Dr. Brad- ley's pamphlet. The needle is a peculiar one, being composed of three or more parallel strips of steel, mounted upon a ring of aluminum, and trimmed to form a circle. Long aluminum pointers are attached to this disk. The needle is balanced upon a steel point, on which rests an agate. The weight of the needle is only twenty grains. " The coils are so placed that the current runs paral- lel with the meridian of the needle. They are half an inch or more wider than the diameter of the disk. By this means all parts of the steel composing the needle are subjected to the same inductive influence in all its deflections. " It is a condition indispensable in the construction of a true tangent galvanometer, that the current through the coil should act as uniformly upon the needle in all its deflections as the earth's magnetism does ; a narrow coil under a long needle does not ful- fill this condition ; for, as the extremities of the needle in its deflections pass more and more away from the coil, the inductive influence is less and less, as com- pared with the earth's influence. 22 TANGENT GALVANOMETER. "On the contrary, if we place a very broad coil un- der a long needle, the same difficulty occurs, but in the opposite direction. While the needle is on the meridian it is under the influence of but few convolu- tions in the middle of the coil, but as it deflects it comes under the influence of an increasing number of convolutions, and therefore the influence is more and more increased. " It being evident that the truth lay between these extremes, the expedient of a needle in the form above described was resorted to, and with entire success, for in this the condition sought is accurately fulfilled. " Coil No. i is composed of very fine copper wire, wound evenly back and forth over the whole width of the coil, and of a sufficient number of layers to give a resistance of 150 or more ohms. " No. 2 is of No. 30 wire wound in the same manner, and to twenty-five or thirty ohms resistance. No. 3 is of two layers of No. 23 wire, giving one to two ohms resistance. And No. 4 is a strip of sheet copper of the width of the coils, and wound three and a half times around, so that the current passes four times under the needle ; the resistance of this may be considered as null, or not sufficient to be noticed or taken into account. " The outer ends of all the coils are connected with a common screw-cup, while the inner ones are con- nected each with its cup bearing its proper number. " Coil No. 1 is for currents of high intensity, No. 4 for those of great quantity, and Nos. 2 and 3 are for mixed or intermediate currents. " The Rheostat contains coils whose several resis- tances range from T -J- 7 of an ohm to 4000 ohms, any one or more of which may be thrown into the circuit by removing the proper plug or plugs on the top of the rheostat, so that any resistance may be introduced from ^-g- of an ohm to 10,000 ohms. " In addition to two screw-cups [I. and II.] for con- TANGENT GALVANOMETER. nection with the battery and galvanometer, there are two other screw-cups [III. and IV.] for connection of any conductor whose resistance it is intended to meas- ure ; also a switch, A, on the rheostat, so arranged that the battery may be thrown through the rheostat or the conductor." CONNECTIONS FOR MEASUREMENT. Connect line or instrument-wires to be measured, to posts III. and IV. 24 TANGENT GALVANOMETER. of the rheostat, one pole of the battery to B, on the galvanometer, and the other to I, on the rheostat. The wire leading from II. of the rheostat, is connected with i, 2, 3, or 4, of the galvanometer, depending upon the coil to be used. Push the switch A to the right, throwing the current through the wire to be measured. Note the deflection of the needle. Now push A to the left-hand plate, throwing the current through the rheostat. Remove plugs, thus letting resistance into circuit, until the needle shows the same, deflection. Add the figures marked on the holes unplugged, and you have your resistance in ohms. When you measure a wire put to ground at distant end, connect as before, except that you put a ground- wire to post IV., and line to III. Manipulate as before. TESTING FOR INSULATION. First unplug 10,000 ohms resistance, using galvanometer coil No. i. Note the deflection with the current through the 10,000 ohms. Call this the " maximum of the galvanometer." Now switch to the line (which is open at the farther end), and note the deflection again. (The better the insulation of line the less the deflection.) Suppose the deflection with the 10,000 ohms of the rheostat to be 30, the tangent of which is .5774*, while that through the line is 10, the tangent of which is .1763. Now the tangent .5774 is to 10,000 ohms, as the tan- gent .1763 inversely is to the answer; or, .5774 x 10,000 hms> .1763 the insulation resistance of the line. This, multiplied by the number of miles, gives the mileage insulation, and this again, by the number of insulators per mile, the average resistance of each insulator. * See table of tangents. GAUGAIN GALVANOMETER. We give the above examples to show the method of using the instrument. Many of the formulae given in this book may be used for various measurements with this instrument. 2. Gamgain Galvanometer. This is also a tan- gent instrument, useful for measuring the electromo- tive force of batteries, their quantity and internal resistance. Its construction is in accordance with the following law : " If a magnetized needle is submitted to the action of a circular current, placed in the magnetic meridian, when the center of the needle occupies the summit of a cone, having for its base the circular current, the tangents of the angles of the deviation of the needle will be nearly proportional to the force of the current, when the height of the cone is equal to one-fourth of the diameter at its base. This theorem is correct within TOTT when the needle is from 1.17 inches to 1.36 inches in length, and the coil not less in diameter than thrice the length of the needle." Fig. 2 (a, b) gives a plan and an elevation of this gal- vanometer. The wire forming the helix is generally wound in two parts of equal length and resistance, thus having FIG. -3 (a). Plan, of Gaugain Galvanometer. 26 DIFFERENTIAL GALVANOMETER. an equal effect upon the needle. Thus this galvano- meter may be used with both coils connected, and the FIG. 2 (3). Elevation of Gaugain Galvanometer. indication given by the deflection as in a tangent in- strument, or the resistance to be measured, may be connected with one-half, and a rheostat with the other, using it as a differential instrument. 3. Differential Galvanometer. In this form the needle is acted upon by two coils of equal length, WHEATSTONE BRIDGE. 27 resistance, and power. The current leads to the coils on one wire, where it divides, passing around the needle in opposite directions. The resistance to be measured is attached to one coil, and the rheostat to the other. When the same amount of resistance is let into circuit in the rheostat, or its equivalent by shunt- ing (see shunt), the effect upon the needle will be the same in each direction, and the needle will be brought to zero. Then the resistance shown in the rheostat (multiplied by the shunt, if one is used,) gives the required information. 4. Wlieatstone Bridge. In tangent and differ- ential galvanometers the effect upon the needle is proportioned to the strength of the current, this being controlled, of course, by the resistance in circuit. The Wheatstone Bridge system of measurement is entirely different, its action being based upon the fall of ten- sion of the current. The fall of tension of a current from the pole of the battery to the wire terminal at the ground is, as we have shown, uniform. If, for a portion of the distance, we use two wires instead of one, the current will di- vide and flow by the two branches, and a point on one wire may be connected to a point on the other by a cross-wire, where the tension of the current is the 28 THE WHEATSTONE BRIDGE. same, without at all affecting the flow of current on the two lines. For, as the flow of current is caused by a difference of tension between the battery and the point with which it is connected by the line-wire, and as the tension of the currents at the two points on the wires connected by the cross-wire is the same, no current will flow through the cross-wire. And a gal- vanometer inserted in this cross-wire will, of course, show no deflection. In Fig. 3, suppose the current to start from E and flow to post i>. Here it divides, one-half passing by wire A, 3, B, to 2, and the rest by C, 4, D, to 2, and thence the current returns to battery. The resistance of the sides, A and C, and B and D, being alike, the tension of the two portions of the current, at 3 and 4, is the same, and no current will pass across, from 3 to 4, through the galvanometer. Again : suppose we insert at A a resistance of 10, at B 100, at C 500, and at D 5000. We shall find that the needle on the cross-wire still refuses to deflect. Be- cause the current divides at"':i, inversely as the resist- ance of the two routes i, A,.3< and i, C, 4; and the resistance at C being 50 times as, great as at A, 50 parts of the current passed through 1 A, with a resistance of 10, while i part passed through C with a resistance of 500. The tensions of the two portions, on arriving at 3 and 4, are the same, and as the same proportion as between A and C, is found between B and D, the divid- ed portions pass on until they join at 2 and return to battery. Therefore, when A bears the same propor- tion to C, that B does to D, or when A : C : : B : D, no current will pass between 3 and 4. Now let us measure an unknown resistance which we will insert between the binding-posts at D. Say we put 20 ohms in at C, and i at A. The needle is deflected strongly. Now insert resistance at B until the needle comes back to zero. Say it requires 250 units. THE WHEATSTONE BRIDGE. 29 Then, 20 x 250 _L_ = 5000, which is the resistance at D. If the resistance to be measured is small, you may insert equal amounts at A and C, and then the number of units inserted at B will exactly equal D. There are two forms of galvanometers in this coun- try, made on the above principle : Siemens' and Gray &* Barton's. The latter we will descif|n& first, as its arrangement is similar to the diagranWjng. 3. The battery-wire is led to a brass/bjMe, on each side of which' are smaller plates connecTSra with resistance coils. By inserting plugs on eagjraitie of the battery- plate, the proportions of resistaflye between the first and second sides of the bridg^JA. and C, Fig. 3) may be varied at pleasure. The ifeflstance to be measured is introduced at D, and a rhecWat box at B. The Siemens' galvanometer is entirely different in appearance, and although/jfct as accurate as the Gray & Barton for small measfljements, is much more con- venient for general lijfejwork, as it is portable, light, and convenient in form. The needle is made astatic. That is, two light nee- dles are affixed rigidly to the same shaft. The two needles are polarized in different directions. One be- ing slightly stronger than the other, the working effect or polarity of the pair is the difference in strength be- tween them. This being very slight, the tendency to remain in the magnetic meridian is very feeble, and an extremely weak battery-current will cause a deflection. To increase still more the sensitiveness of the needle, it is suspended by a single fibre of untwisted cocoon silk. The lower needle hangs inside the coil, while the upper, which serves also for an indicator, hangs across the top of the dial. The dial and needle are covered with a glass case ; surrounding the base of the glass 30 THE WHEATSTONE BRIDGE, horizontally is a slate disc, divided into 300 degrees, running 150 from each side of the centre or o mark. Half embedded in the edge of this slate is a platinum or German-silver wire. This wire forms the first and second sides of the Wheatstone bridge the A and C of Fig. 3. Pressing against this wire is a small platinum wheel, with a vernier, on a movable arm. To this the battery- wire is connected. In the base of the instrument are three resistance-coils, 10, 100, and 1,000, either one of which is let into the third side (B, Fig. 3) by removing the plug which cuts it out. The resistance to be meas- ured is introduced into side 4 (D, Fig. 3). The principle of measurement is precisely like that shown in Fig. 3 ; but in the Siemens', sides B and D being fixed, you vary the proportion between A and C until the proportion of the latter two is the same as the first two. The slate being divided into two parts by the divi- sions of 150 each, one side is marked A and the other B. Fig. 4 will show the connections. To measure the resistance of a relay, for instance, we connect the bat- tery to posts I. and II., and the wires from the resist- ance to be measured to posts II. and IV. The plug re- mains in, between III. and IV. Remove the plug at 100. The current passes from post I. to roller Z. There it divides, one portion going by the wire on the edge of the slate, through the resistance coil 100 back to battery, by post II. The other portion of the current passes (on the B side of the slate) to posts III. and IV., thence out through the relay to be measured, and back to battery at post II. Remember, that the wire between the roller Z and 150, on one side of the slate, is the first side (A, Fig. 3) ; that between the roller and 1 50 on the opposite side corre- sponds with the second side (C, Fig. 3) ; the resistance 100 to post II., the third side (B, Fig. 3) ; and from post IV., through the relay to be measured, back to post II. (D, THE WHEATSTONE BRIDGE. Fig. 3), is the fourth side ; while from C, through the galvanometer, is the cross or bridge-wire. FIG. 4. Now, when by moving the roller Z to the left or right, the proportion between the sides of the slate, A and B, is the same as between the resistance 100 and the relay to be measured, the needle will come to zero. Suppose that when the needle stands at o, the wheel Z is at 50, on the A side of the slate. Then, as A is to B, so is 100 to X, the unknown resistance. There are 1 50 degrees on each half of the slate. Then the for- mula would read, 1 50 + 50 x 100 = X, or 15050 100 which is the resistance sought. x loo = 200 , 32 SINE GALVANOMETER. When a line is to be measured, connect battery to post I., ground to II., and line to IV. The ground on post IJrcompletes the circuits, as the line is grounded at distant end, and the other end of the battery is also grounded. 5. Sine Galvanometer. The strength of the cur- rent, in a sine galvanometer, is proportional to the sine of the angle of deflection. The Siemens' may be used as a sine galvanometer by connecting the battery to post II., and line to be measured to IV., and unplug- ging between III. and IV. Set the needle in position ; put the vernier-wheel at o, on the slate. Now, when the needle is deflected, move the needle-coil by revolv- ing the slate in the direction of the deflection of the needle. The coil is thus kept parallel with the needle, and its maximum effect exerted. When the needle is deflected to that point where the earth's attraction for the needle is exactly balanced by the effort of the coil- current to deflect it, the needle will become stationary. Now read the degree on the slate opposite the vernier, and a reference to the table of sines will give the sine of the deflection ; and having previously taken the constant of your instrument that is, having found to what sine it will deflect with a given resistance and a given battery you have now a proportion giving you the result sought. FORMULAE. 33 CHAPTER III. FORMULA: TO MEASURE ELECTROMOTIVE FORCE AND INTERNAL RESIST- ANCE OF A BATTERY. 1st. By Method of Equal Deflections. The standard cell being .............................. E, The one to be compared being ................... E'. Take the deflection of E and call it ............... R, Then take the deflection of E' and call it ......... R', Adding resistance to make the deflection the same. Then, 2d. By Direct Measurement, giving also the internal resistance of the battery (with Bradley or Gaugain). First, take tangent of deflection, without resist- ance, call it ............................... I Then take it with given resistance ............. = I' Let the resistance used be called ............... R The resistance of galvanometer ................ = G And of the battery ............................ X . For the electromotive force, multiply 7by^+, or fx(X+G) = E. And the internal resistance of the battery is found 34 INTERNAL RESISTANCE OF BATTERY. by subtracting the resistance of the galvanometer from 7 2 x R Example. Suppose the resistance of the galvano- meter to be .56 of an ohm. The first deflection is 80, of which the tangent is 5.671 With 10 ohms resistance it is 52, of which the tangent is ................................. i .280 5.671 1.280 = 4.391 . Multiply the second tangent, 1.280, by the resistance, 10, and we have 12,800, which, divided by the differ- ence between the two tangents, 12,800 , =2.96, which is the total resistance in circuit. Deduct the galvanometer resistance, .56, and we have the internal resistance of the battery, = 2.40. The tangent of I was 5.671, and this, multiplied by the total resistance in circuit, gives 16.78616 as the electromotive force of the battery. Internal Resistance of Battery. Put the bat- tery in circuit with a sine or tangent galvanometer. Note the deflection. Halve the tangent of the deflec- tion by introducing resistance. The resistance intro- duced is equal to the original resistance that of the battery and the galvanometer-coil. Deduct the latter and you have the desired result. Mance's Formula, A and B are two resistances MEASUREMENT OF INSULATION. 35 whose ratio is known. R, an adjustable rheostat ; E, battery. FIG. 4. When the needle is balanced, by closing wire C D atK, E-R A 2?' This is regardless of the resistance of the galvano- meter. The Wheatstone bridge may be used for this mode of measurement. Sir Wm. Thomson's modification of the above gJ^es the resistance of the galvanometer by the deflection of its own needle. Thus, FIG. 5. Balance made as before. 36 MEASUREMENT OF ELECTROMOTIVE FORCE. Now the resistance of the galvanometer, G, is, G = R -B- Measurement of Electromotive Force by Comparison. (Clark.) When a number of cells are joined up in circuit with, but in opposition to, a number of other cells, with a galvanometer inserted, by adjusting the number of cells so that no current passes, the relative electromotive force of the two bat- teries may be determined. Or, call the electromotive forces of the two batteries E and E' ; join them up successively in circuit with the same galvanometer, and by varying the resistance cause them both to give the same deflection ; their forces will then be in direct proportion to the total resistances in circuit in each case, or, When R represents the resistance with E (including that of battery, galvanometer, and the adjustable resist- ance) and R' with E'. Measurement of Insulation. The table given on page 7, of the specific resistances of metals, might be indefinitely extended, embracing liquids of all kinds, gums, resins, glass, minerals, etc. Silver and copper, the first on the list, represent the best con- ductors, while dry air ranks lowest as a conductor, and therefore highest as an insulator, insulation being the converse of conduction. The insulating power of any material with reference to a given current depends upon the tension or energy of that current. The less the tension, the less power the current has to overcome the resistance of a con- ductor; hence water, which is a fair conductor of the currents used on ordinary telegraph lines, is an ex- cellent insulator for currents of low tension, and dry TESTING FOR INSULATION. 37- glass, which is an insulator for telegraphic currents may be perforated by currents of sufficient tension while atmospheric electricity readily overcomes the resistance of dry air. Insulation is, therefore, a rela- tive term. Now, glass, porcelain, etc., are absolute non-conductors of telegraphic currents. Theoreti- cally, therefore, these substances should be perfect insulators for telegraphic currents. In practice, how- ever, glass, and in a less degree porcelain, part with heat when cooled ; hence a thin film of water con- densed from the moisture of the air forms upon the surface of each insulator, and conducts electricity from the wire to the supports. Upon a long line the total leakage from the great number of insulators is therefore very sensible. Accumulations of dust and smoke upon the surface of an insulator absorb and retain moisture. A similar effect arises from the webs of insects connecting the outside of the insulator with its support. In order to prevent the escape from the direct ac- tion of rain, a flange or petticoat is formed upon the insulator, giving it a concave under-surface. During a rainfall this is the only serviceable portion of the insu- lator, as the exterior is covered with a film of water. Thus the aggregate escape in moist weather is consi- derable, even when every insulator is sound ; broken insulators, of course, greatly increase the leakage. Measurements for general or average insulation are, of course, valueless when there is a great leakage at one point. If the line is opened at the distant end, and the deflection of the galvanometer shows that a current still flows to line, it is obvious that it must pass to ground over the surface of the insulators. If the escape is practically uniform along the line it may be measured thus: 09 38 TESTING FOR CONDUCTIVITY LINE RESISTANCE. Open line at distant end, connect your battery to line throug) the galvanometer ; the resistance meas- ured and calculated in the usual manner will be the insulation-resistance of the entire line, which multi- plied by its length in miles gives the mean insulation- resistance per mile, and again multiplied by the num- ber of poles per mile gives the mean resistance per insulator. Explanation. To ascertain wire-resistance, divide the total resistance by the length in miles, because the resistance increases in proportion to the length. To ascertain insulation-resistance per mile multiply the total resistance by the mileage, because the es- cape increases in proportion to the length, two miles giving twice the escape, and hence one-half the insu- lation of one mile. Testing for Conductivity. First, test with line open at distant end, for if there is a heavy escape on the line your resistance will show less than it really is. Then close the wire, and the result will be the entire resistance of the wire and instruments (if they are in circuit). To test the wire carefully, the relays should all be cut out. The total resistance, divided by the number of miles, gives the wire resistance per mile. Line Resistance Measured from an Interme- diate Station. Mr. F. L. Pope gives the following ingenious method of measurement with the Bradley Tangent, putting it into circuit anywhere on the line. Any good tangent could be used in the same manner: " Connect the main circuit so that it will pass through I. and II. of the rheostat, and through the No. i coil of the galvanometer. When the circuit is closed for a few seconds take a reading of the deflection, then unplug a sufficient resistance to reduce the deflection perhaps one-half (1,000 or 2,000 ohms will usually an- LINE RESISTANCE. 39 swer), and take a second reading. It is well to reverse the connections and repeat the two observations, and take the mean result of the two deflections. The fol- lowing example will serve to show how the result is worked out : " ' Suppose the deflection obtained on No. i coil in the main circuit is 61 (tan. 1.804), and by inserting 2,000 ohms it is reduced to 42 (tan. .900). If we call the unknown resistance x, we have, x : x 4- 2000 : : .900 (tan. 42) : 1.804 (tan. 61)- Converting this into an equation and reducing, .904.3: = 1800000 x = 1991.' " Or (to avoid equations), subtract the tan. of 42 (.900) from tan. 61 (1.804) and we have .904 for the equivalent tan. for the 2000 ohms inserted. Now, " Tan. .904 : 2000 ohms : : tan. .900 : 1991 ohms. "The resistance of all relays in the line are, of course, to be subtracted." Corrected One Resistance by Schwendler. This is only applicable to a line having a general escape its entire length. In such case you obtain the absolute line resistance, irrespective of the escape. Let, R = measured insulation resistance of line. r = wire resistance without relay at distant end. r' wire resistance including relay at distant end. Then, [i] The Corrected Line-wire resistance L, is, Z = 2(7? 40 LINE RESISTANCE TESTING FOR GROUNDS. [2] The Corrected Insulation resistance R, is, R = y ( - r ) . [3] The Relay resistance r', is, R(r'-r) R-r 1 ' SECOND METHOD. R = apparent insulation resistance. r = apparent conductivity. The corrected line resistance L, will be, This is based on the supposition that the entire escape is concentrated at one point midway of the cir- cuit. Unless the line is very long, and the escape ex- tremely bad, this formula is very accurate. And on the same supposition the corrected insulation resist- ance, R', will be, R' = R--. 2 Testing* for Grounds. ist. FROM EACH END. Having a record of the mileage resistance of the wire, let it be measured from each end to the fault. Call the registered or known resistance of the wire R. Measurement from the most distant end through the fault to ground (other end open) L. Ditto from the opposite end S. r> Then L + S = G , the resistance of the fault ; L G gives the distance of long end, and S G distance of short end from fault. TESTING FOR GROUNDS. 41 2d. BY LOOP TEST. With Differential Galvanome- ter. Connect two wires in metallic loop at distant end. Measure the loop, disregarding the fault. Now con- nect the two wires to the two circuits of the galvano- meter, attaching- battery with one end to ground. As one wire runs to the station beyond the fault, and re- turns to the fault to take ground, it will show the most resistance. Put resistance in the other side to balance. Call this resistance b ; call resistance of metallic loop a. Then = distance from measuring station on shortest wire to fault. With Tangent Galvanometer. ist. Measure loop, which is a. 2d. Take deflection on longest wire, with shortest open at your instrument. 3d. Take deflection of shortest wire, with longest open. Now add resistance to make third measurement equal the second. Call this resistance b. And as before, = distance on short wire. With Siemens ', without Rheostat. (Haskms.) Call resistance of metallic loop a. Call measurement of longest part of loop b. Call measurement of shortest part of loop c. Call resistance of fault d. And distance of fault from office x. b + c a _ = d, and c d x or, b cg, the difference between the two legs of the loop ; and, 42 TESTING FOR CROSSES. 3d. FROM ONE END. (Blavier.) Call the registered or known resistance of the wire R. Measure with further end closed (current escaping - also by the fault) S. Then measure open at distant end. Call this T. Call distance of fault X. And resistance of fault. . . Z. ^=5 OO l^. t~> ONMVONON ONO^Ooo ON ro O O ^ slim niii nm imi rh p vo ^vo &% ScS H H . H it ii mil ii til mi! W vo CO ro N ON N 00 00 HI tovo vo t^. ON O> HI ro to t^. 5 H ._ i 3 M nf M ^ ^. 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QCM'^-t^.o CM moo o ro ^ininmm vONOvovqt^. t->t^. t^oo oo i cs m g m ^S^ONC c?^ O 1 H* * ON rs-vo^oo VO t^oo ONO Hcsro-^io vo t-oo ON O w 10 m to o n NATURAL TANGENTS. * m mvo tx, c-x oo ON o HI N tx O vo ro ro M- tx, mvo co ^t-vo ON roco ro * m rxoo w N cl N O 1000 VO tx, m m m HI HI ON ^vo M m mvo o oo O N ^t- rx q> W in O ON N ONVO ON HI Tf W vo 5 vo N x m invo rx rx co ON o HI N M Tf M ONVO ro o^co^ rx M^ N 00 N H\O N co rx rx r> vo N >n m rx vo O m M j ^ inn mil ON r?co~ rom N rovo O m ro TT 10 i-xco ro rooo vo txVO tx, fO HJ ro O ON m ro Iff!! 1 """ W N N N ro ro ro ro ro ^t- "* m mvo 00 mvo VO M VO O ON O ON vo oo o N m M O M 00 - ^ ft -*-vo H W HI m rx N oo HI ON tx ON <* rO VO ifOO rx * ON m M oo m N 6 oo rx rx rx ON n ^t ON N ro mvo tx >i- ro ro rx M ON M ro moo M roco ro M O ro'w ON m HI rt-cO M OO . H * if mvo vo rx co co ON o H O H. 00 N w ro ON HI lovo vo C rx o ro f^ N ro mvo rx CO ^vo ro to ro " M^ rooT rx N rooo oo H M'VO" N ON tx. QCO ^-00 O *- tx N rx 5 vo vo rxoo o O tx HI N N ro roco -t- rx vo vo ON rOOO M ro <* ro f 1 %SHr? cSl^l?? vo tx txoo N invo oo M vo |l?|l o" ro tx T fx vo rx roco 10 to ro M O HI III! HI co m M in CM vo 00 -* O O tx rxoo ro ON O N "* rx O HJ -4- ^- M HI ro O * tx O ro rx M vo 5 *b vor.ooON^ H ro^io -0-00-0 M fr. ^. 10 vo rxoo ON O Q ' NATURAL TANGENTS. OOflNON * N O ON H N " o lllll t!% w 43 M | * IS H., ^ C c 8 'H -rt CJ CO OO f- p COOO ON 1 S ON 8 s 00 n CO VO CO ON ? m G co NO tx. ON O rp *V ^ J fl *N o CO CO ri il II II vo t^-od ON vo ro 6 ro I k, ^O) 1 O "t- N f OO -^-MOON fi ~> V) vo txoo o M in N in 10 MM M W CO ON 3 S m\o w oo ro ro * co tx. i fen 1 %) OT V) /o c^ & 'O g 75 g S 8^ VO Hill mi S ^ Q 1 o 'G 'O ON NO tx,oo o> M in o M M S S; ON ON ro o 00 ^ VO txOO ON M <* ON O rr 1 o II II q" CO II ON II J4 II R||S ^fp, g u S Q ^ s ?w*J S|2^ M N ro ^- it) VO hxOO ON Q 0000000000 CO 00 CO OO ON SINES AND TANGENTS. Table of Sines and Tangents. Deg. Sine. Tangent. Deg. Sine. Tangent. 0.5 00873 00872 29 48481 55430 i 01745 01745 29-5 49242 56577 i-5 02618 02618 30 50000 57735 2 03490 03492 30.5 50754 58904 2.5 04362 04366 31 51504 60086 3 05234 05240 31-5 52250 61280 3-5 06105 06116 32 52992 62486 4 06796 06992 32 5 53730 63707 4-5 07846 07870 33 54464 64940 5 08716 08748 33-5 55194 66188 5-5 09585 09628 34 55919 67450 6 i453 10510 34-5 56641 68728 6-5 11320 "393 35 57358 70020 7 12187 12278 35-5 58070 71321 7-5 13053 I3 l6 5 36 58779 72654 8 i39 J 7 14054 36.5 59482 73996 8-5 14781 14945 37 60182 75355 9 15643 ^5838 37-5 60876 76732 9-5 16505 l6 734 38 61566 78128 10 17365 17632 38.5 62251 79543 10.5 18224 i8533 39 62932 80978 ii 19081 19438 39-5 63608 82433 "5 19937 20345 40 64279 83910 12 20791 21255 40-5 64945 85408 12.5 21644 22169 4i 6^606 86928 *3 22945 23086 4i-5 66262 88472 J 3-5 23345 24007 42 66913 90040 *4 24192 24932 42-5 67559 9 l6 33 14.5 23038 25861 43 68200 93251 15 25882 26794 43-5 688^5 15-5 26724 27732 44 69466 96568 16 27564 28674 44-5 70091 98269 16.5 28402 29621 45 70711 .00000 17 29237 30573 45-5 7*3S .01761 17-5 30071 31529 46 7*934 03553 18 30902 32492 46.5 7^537 .05378 18.5 3^30 33459 47 73 r 35 .07237 19 3 2 557 34432 47-5 73728 09131 19-5 3338i 354i * 48 74314 .11061 20 34202 36397 48.5 74896 . 13029 20.5 352i 37388 49 75471 15037 21 ~ "35*37 38386 49-5 76041 .17085 21.5 36650 " 3$39*W.,.\ go 76604 I 9 I 75 22 3746i 40402 *j-"-. \ 5*5 77162 .21310 22.5 38268 41421 5i 777]5 .23490 23 3973 42447 5i-5 '78261 .2-717 23-5 39875 4348i 52 78801 .27994 24 40674 44522 52.5 79335 .30323 24-5 41469 45572 53 79864 32704 25 42262 46630 53-5 80386 35H2 25-5 43051 47697 54 80902 37638 26 43837 48773 54-5 81412 40195 26.5 44620 49858 55 81915 .42815 27 27.5 45399 46i75 50952 52056 55 5 56 82413 82904 .45501 .48256 28 38.5 46947 47716 53 * 7 54295 56.5 57 5 .51084 .53987 SINES AND TANGENTS. 55 Table of Sines and Tangents. Concluded. Deg. Sine. Tangent. Deg. Sine. Tangent. 57-5 84339 56969 74 96126 3.48741 58 84805 .60033 74-5 96363 3.60588 58.5 59 85264 85717 .63185 .66428 75 75-5 96593 96815 3-73205 3.86671 59-5 86163 .69766 76 97030 4.01078 60 86603 73205 76.5 9/237 4.16530 60.5 87036 .76749 77 97437 4-33H8 61 87462 .80405 77-5 97630 4-51071 61.5 87882 .84177 78 97815 4 . 70463 62 88295 .88073 78.5 97992 4.91516 62.5 88701 .92098 79 98163 5-14455 63 89101 .96261 79-5 98325 5-39552 63-5 64 64-5 89493 89879 90259 .00569 . 05030 .09654 80 80.5 81 98481 90629 98769 5.67128 5-97S7 6 6.3*375 65 90361 81.5 98902 6.69116 65.5 90996 1943 82 99027 7- IJ 537 66 66.5 91355 91706 .24604 .29984 82.5 83 99144 99255 7-59575 8.14435 67 92050 35585 83-5 99357 8.77689 67-5 92388 .41421 84 99452 9.5I43 6 68 92718 .47509 84.5 9954 10.3854 68.5 & 69-5 93042 93358 93367 53865 .60509 .67462 85 85-5 86 99619 99692 99756 11.4301 12 . 7062 14 . 3007 70 93969 74748 86.5 16.3409 70.5 94264 . 82391 87 99863 94552 94832 .90421 .98869 87-5 88 99905 99939 22.9038 28 . 6363 72 95106 3-07768 88.5 99966 38.1885 72.5 73 95372 3-i7i59 3.27085 89 89-5 99985 99996 57.7900 II 4 589 73-5 95882 3-37594 90 lOOOOO ~~~ 56 bIZE, WEIGHT, KESISTANCE, ETC., OF IRON WIRE. Table of the Sizes, Weights, Resistances, and Breaki&ig Strain of Iron Wires. No. of wire, B. W. G. Diam. in mils, B. W. G. PER STATUTE MILE. Breaking strain, soft wire. American gauge diameter. Weight in Ibs. Resistance in ohms. i sq. inch, i circ. u oooo 1 7,645 13^58 2,854 0.340 o-433 2.10 1000 454 460 ooo 00 .425 .380 340 2,502 2,001 1, 6oO 2.40 3 o 3-74 8600"' 7100 .40964 .36480 .32495 I 2 3 .300 .284 259 1,245 1,117 928 4.81 5-37 6.46 6000 4850 4000 .28930 25763 .22942 4 .238 .220 .203 783 670 570 7-65 8.96 10.52 3400 2950 2500 .20431 . 18194 . 16202 I 9 .l8o .165 .148 44 8 376 303 13-38 16.39 19.79 22OO 1750 1500 .14428 .12849 .IJ443 10 ii 12 134 .I2O .109 249 199 l6 4 24.14 30.10 36.49 I20O 820 7 IO .10189 .09074 .08081 The Birmingham wire-gauge is really not one gauge, but an approximation only, as different manufacturers do not agree in sizes. The American gauge is formed upon a geometrical progression. The last column of the above table gives 'the American gauge diameter of the various numbers. (For the figures in this col- umn we are indebted to Pope's Modern Practice, page 146.) The columns of weight, breaking-strain, and resistance, all refer to wire of Birmingham gauge. TABLE OF SQUARES, CUBES, SQUARE AND CUBE KOOTS, Number. Squares. Cubes. s/Koots. v^ Roots. Reciprocals. 1 1 1 1-0000000 1-0000000 l.ooooooot 2 4 8 1-4142136 1-2599210 500000000 3 9 27 1-7320508 1-4422496 333333333 4 16 64 2-0000000 1-5874011 250000000 5 25 125 2-2360680 1-7099759 200000000 6 36 216 2-4494897 1-8171206 166666667 7 49 343 2-6457513 1-9129312 142857143 8 64 512 2-8284271 2-0000000 125000000 9 81 729 3-0000000 2-0800837 111111111 10 100 1000 3-1622777 2-1544347 100000000 11 121 1331 3-3166248 2-2239801 090909091 12 144 1728 3-4641016 2-2894286 083333333 13 169 2197 3-6055513 2-3513347 076923077 14 196 2744 3-7416574 2-4101422 071428571 15 225 3375 3-8729833 2-4662121 066666667 16 256 4096 4-0000000 2-5198421 062500000 17 289 4913 4-1231056 2-5712816 058823529 18 324 5832 4-2426407 2-6207414 055555556 19 361 6859 4-3588989 2-6684016 052631579 20 400 8000 4-4721360 2-7144177 050000000 21 441 9261 4-5825757 2-7589243 047619048 22 484 10648 4-6904158 2-8020393 045454545 23 529 12167 4-7958315 2-8438670 043478261 24 576 13824 4-8989795 2-8844991 041666667 25 625 15625 5-0000000 2-9240177 040000000 26 676 17576 5-0990195 2-9624960 038461538 27 729 19683 5-1961524 3-0000000 037037037 28 784 21952 5-2915026 3-0365889 035714286 29 841 24389 5-3851648 3-0723168 034482759 30 900 27000 5-4772256 3-1072325 033333333 31 961 29791 5-5677644 3-1413806 032258065 32 1024 32768 5-6568542 3-1748021 031250000 33 1089 35937 5-7445626 3-2075343 030303030 34 1156 39304 5-8309519 3-2396118 029411765 35 1225 42875 5-9160798 3-2710663 028571429 36 1296 46656 6-0000000 3-3019272 027777778 17 1369 50653 6-0827625 3-3322218 027027027 38 1444 54872 6-1644140 3-3619754 026315789 39 1521 59319 6-2449980 3-3912114 025641026 40 1600 64000 6-3245553 3-4199519 025000000 41 1681 68921 6-4031242 3-4482172 024390244 42 1764 74088 6-4807407 3-4760266 023809524 43 1849 79507 6-5574385 3-5033981 023255814 44 1936 85184 6-6332496 3-5303483 022727273 45 2025 91125 6-7082039 3-5568933 022222222 46 2116 97336 6-7823300 3-5830479 021739130 47 2209 103823 6-8556546 3-6088261 021276600 48 2304 110592 6-9282032 3-6342411 020833333 49 2401 117649 7-0000000 3-6593057 020408163 50 2500 125000 7-0710678 3-6840314 020000000 51 2601 132651 7-1414284 3-7084298 019607843 62 2704 140608 7-2111026 3-7325111 019230769 TAELS or SQUARES, CUBES, SQUARE AND CUBE ROOTS. Number. Squares. Cubes. /Koots. 4/JRoots. Reciprocals. 469 219961 103161709 21-6564078 7-7C94620 002132196 470 220900 103823000 21-6794834 7-7749801 002127060 471 221841 104487111 21-7025344 7-7804904 002123142 472 222784 105154048 21-7255610 7-7859928 002118644 473 223729 105828817 21-7485632 7-7914875 002114165 474 224C76 106496424 21-7715411 7-7969745 002109705 475 225625 107171875 21-7944947 7-8024538 002105263 476 226576 107850176 21-8174242 7-8079254 002100840 477 227529 108531333 21-8403297 7-8133892 002096486 478 228484 109215352 21-8632111 7-8188456 002092050 479 229441 109902239 21-8860686 7-8242942 002087683 480 230400 110592000 21-9089023 7-8297353 002083333 481 231361 111284641 21-9317122 7-8351688 002079002 482 232324 111980168 21-9544984 7-8405949 002074689 483 233289 112678587 21-9772610 7-8460134 002070393 484 234256 113379904 22-0000000 7-8514244 00206C116 485 235225 114084125 22-0227155 7-8568281 002061856 486 23C196 114791256 22-0454077 7-8622242 002057613 487 237169 115501303 22-0680765 7-8676130 002053388 483 238144 116214272 22-0907220 7-8729944 002049180 489 239121 116930169 22-1133444 7-8783684 002044990 490 240100 117649000 22-1359436 7-8837352 002040816 491 241081 118370771 22-1585198 7-8890946 0020366GO 492 242064 119095488 22-1810730 7-89444C8 002032520 493 243049 119823157 22-2036033 7-8997917 002028398 494 244036 120553784 22-2261108 7-9051294 002024291 495 245025 121287375 22-2485955 7-9104599 002020202 496 246016 122023936 22-2710575 7-9157832 00201C129 497 247009 122763473 22-2934908 7-9210994 002012072 498 248004 123505992 22-31591C6 7-9264085 002008032 499 249001 124251499 22-3383079 7-9317104 002004008 500 250000 125000000 22-3606798 7-9370053 002000000 501 251001 125751501 22-3830293 7-9422931 00199C008 502 252004 126506008 22-4053505 7-9475739 001992082 503 253009 127263527 22-4276615 7-9528477 0019SS072 504 254016 128024064 22-4499443 7-9581144 001984127 505 255025 128787625 22-4722051 7-963C743 001980198 506 256036 129554216 2-4944438 7-9GS6271 001976285 507 257049 130323843 22-5166605 7-9738731 001972387 503 258064 131096512 22-5388553 7-9791122 001C68504 509 259081 131972229 22-5610283 7-9843444 001964637 510 260100! 132651000 22-5831796 7-9895697 0019C07S4 511 261121 133432831 22-6053091 7-99478S3 001956947 512 262144 134217728 22-6274170 8-0000000 001953125 513 263169 135005697 22-6495033 8-0052049 001949518 514 264196 i 135796744 22-6715681 8-0104032 001945525 515 265225 136590875 22-6936114 8-0155946 001941748 516 266256 137388096 22-7156334 8-0207794 001937t)84 517 267289 , 138188413 22-7376341 8-0259574 001934236 518 268324 138991832 22-7596134 8-0311287 001930502 519 269361 139798359 22-7815715 8-0362935 001926782 520 I 270400 140G08000 22-8035085 , 8-0414515 001923077 TABLE OP SQUARES, CBEES, SQUARE AND CUBE HOOTS. Number. Squares. Cubes. VKootg. 4/ Roots. Reciprocals. 521 271411 141420761 22-8254244 8-0466030 001919386 522 272484 142236648 22-8473193 8-0517479 001915700 523 273529 143055667 22-8691933 8-0568862 001912046 521 274576 143877824 22-8910463 8-0620180 001908397 525 275625 144703125 22-9128785 8-0671432 001904762 526 276676 145531576 22-9346899 8-0722620 001901141 527 277729 146363183 22-9564806 8-0773743 001897533 523 278784 147197952 22-9782506 8-0824800 001893939 529 279841 148035889 23-0000000 8-08/5794 001890359 530 280900 148877001 23-0217289 8-0926723 001886792 531 281961 149721291 23-0434372 8-0977589 001883239 532 283024 150568768 23-0651252 8-1028390 001879699 533 284089 151419437 23-0867928 8-1079128 001876173 534 285156 152273304 23-1084400 8-1129803 001872659 535 286225 153130375 23-1300670 8-1180414 001869159 536 287296 153990656 23-1516738 8-1230962 001865672 537 288369 154854153 23-1732605 8-1281447 001862197 538 289444 155720872 23-1948270 8-1331870 001858736 539 290521 156590819 23-2163735 8-1382230 001855288 540 291600 157464000 23-2379001 8-1432529 001851852 541 292681 158340421 23-2594067 8-1482765 001848429 542 293764 159220088 23-2808935 8-1532939 001845018 543 294849 160103007 23-3023604 8-1583051 001841621 544 295936 160989184 23-3238076 8-1633102 001838235 545 297025 161878625 23-3452351 8-1683092 001834862 546 298116 162771336 23-3666429 8-1733020 001831502 547 299209 163667323 23-3880311 8-1782888 001828154 543 300304 164566592 23-4093998 8-1832695 001824818 549 301401 165469149 23-4307490 8-1882441 001821494 550 302500 166375000 23-4520788 8-1932127 001818182 551 303601 167284151 23-4733892 8-1981753 001814882 552 304704 168196608 23-4946802 8-2031319 001811594 553 305809 169112377 23-5154)520 8-2080825 001808318 554 306916 170031464 23-5372046 8-2130271 001805054 555 308025 170953875 23-5584380 8-2179657 001801802 556 309136 171879616 23-5796522 8-2228985 001798561 557 310249 172808693 23-6008474 8-2278254 001795332 553 311364 173741112 23-6220236 8-2327463 001792115 559 312481 174676879 23-6431808 8-2376614 001788909 5GO 313600 175616000 23-6643191 8-2425706 001785714 5G1 314721 176558481 23-6854386 8-2474740 001782531 5G2 315844 177504328 23-7065392 8-2523715 001779359 5G3 316969 178453547 23-7276210 8-2572635 001776199 564 318096 179406144 23-7486842 8-2621492 001773050 565 319225 180362125 23-7697286 8-2670294 001769912 566 320356 181321496 23-7907545 8-2719039 001766784 567 321489 182284263 23-8117618 8-2767726 001763668 563 322624 183250432 23-8327506 8-2816255 001760563 569 323761 184220009 23-8537209 8-2864928 001757469 570 324900 185193000 23-8746728 8-2913444 001754386 571 326041 186169411 23-8956063 8-2961903 001751313 572 327184 187149248 23-9165215 8-3010304 001748251 TABLE OP SQUARES, CUBES, SQUARE AND CUBE ROOTS. Number. Squares. Cubes. 4 440896 292754944 25-7681975 8-7241414 001506024 665 442225 294079625 25-7875939 8-7285187 001503759 666 443556 295408296 25-8069758 8-7328918 001501502 667 444899 296740963 25-8263431 8-7372604 001499250 668 446224 298077632 25-8456960 8-7416246 001497006 669 447561 299418309 25-8650343 8-7459846 001494768 ero 448900 300763000 25-8843582 8-7503401 001492537 671 450241 302111711 25-9036677 8-7546913 001490313 672 451584 303464448 25-9229628 8-7590383 001488095 673 452929 304821217 25-9422435 8-7633809 001485884 674 454276 306182024 25-9615100 8-7C771S2 001483680 675 455625 307546875 25-9807621 8-7720532 001481481 676 456976 308915776 26-0000000 8-7763830 001479290 TABLE OP SQUARES, CUBES, SQUARE AM> CUBB ROOTS. Number. Squares. Cubes. \f Boots. % Roots. Reciprocal 677 458329 310288733 26-0192237 8-7807084 0014771 C7S 459684 311665752 26-0384331 8-7850296 0014749 679 461041 313046839 26-0576284 8-7893406 0014727 680 4G2400 314432000 26-0768096 8-7936593 0014705 651 4G3761 315821241 26-0959767 8-7979679 0014684 CS2 465124 317214568 26-1151297 8-8022721 0014662 683 466489 318611987 26-1342687 8-8065722 0014641 6S4 467856 320013504 26-1533937 8-810S6S1 0014619 685 469225 321419125 26-1725047 8-8151598 0014598 6S6 470596 322828856 26-1916017 8-8194474 0014577 637 471969 324242703 26-2106848 8-8237307 0014556 688 473344 325660672 26-2297541 8-8280099 0014534 6S9 474721 327082769 26-2488095 8-8322850 0014513 690 476100 328509000 26-2678511 8-8365559 0014492 691 477481 329939371 26-2868789 8-8408227 0014471 692 478864 331373888 26-3058929 8-8450854 0014450 693 4S0249 332812557 26-3248932 8-8493440 0014430 694 481636 334255384 26-3438797 8-8535985 0014409 C95 483025 335702375 26-3628527 8-8578489 0014388 696 484416 337153536 26-3818119 8-8620952 0014367 697 485809 338608873 26-4007576 8-8663375 0014347 693 487204 340068392 26-4196896 8-8705757 0014326 609 488601 341532099 26-4386081 8-8748099 0014306 700 490000 343000000 26-4575131 8-8790400 0014285 701 491401 344472101 26-4764046 8-8832661 0014265 702 492804 345948408 26-4952826 8-8874882 0014245 703 494209 347428927 26-5141472 8-8917063 0014224 704 495C16 348913664 26-5329983 8-8959204 0014204 705 497025 350402625 26-5518361 8-9001304 0014184 706 498436 351895816 26-5706605 8-9043366 0014164 707 499849 353393243 26-5894716 8-90853S7 0014144 708 501264 354894912 26-6082694 8-9127369 0014124 709 502681 356400829 26-6270539 8-9169311 0014104 710 504100 357911000 26-6458252 8-9211214 0014084 711 505521 359425431 26-6645833 8-9253078 0014064 712 506944 360944128 26-6833281 8-9294902 0014044 ris 508369 362467097 26-7020598 8-9336687 0014025 714 509796 363994344 26-7207784 8-9378433 0014005 715 511225 365525875 26-7394839 8-9420140 0013986 710 512656 367061696 26-7581763 8-9461809 0013966 717 514089 368601813 26-7768557 8-9503438 0013947 718 515524 370146232 26-7955220 8-9545029 0013927 719 516961 371694959 26-8141754 8-9586581 0013908 720 518400 373248000 26-8328157 8-9628095 0013888 721 519841 374805361 26-8514432 8-9669570 0013869 722 521284 376367048 26-8700577 8-9711007 0013850 723 522729 377933067 26-8886593 8-9752406 0013531 724 524176 379503424 26-9072481 8-9793766 001381? 725 525625 381078125 26-9258240 8-9835089 0013793 726 527076 382657176 26-9443872 8-9876373 0013774 ,727 528529 384240583 26-9629375 8-9917620 0013755 923 629984 385828352 26-9814751 8-9958899 0013756 TABLE OP SQUARES, CUBES, SQUARE AND CUBE ROOTS. amber. Squares. Cubes. V/ Roots. j/ Roots. Reciprocz 729 531441 387420489 27-0000000 9*0000000 -001371 730 532900 389017000 27-0185122 9*0041134 001369 731 534361 390617891 27-0370117 9*0082229 001367 732 535824 392223168 27*0554985 9*0123288 001366 733 537289 393832837 27- '.'739727 9*0164309 001364 734 538756 395446904 27-0924344 9-0205293 001362 735 540225 397065375 27*1108834 9-0246239 001360 736 541696 398688256 27-1293199 9-0287149 001358 737 543169 400315553 27-1477149 9-0328021 001356 738 544644 401947272 27-1661554 9-0368857 001355 739 546121 403583419 27-1845544 9-0409655 001353 740 547600 405224000 27-2029140 9-0450419 001351 741 549801 406869021 27-2213152 9*0491142 001349 V42 550564 408518488 27-2396769 9*0531831 001347 743 552049 410172407 27-2580263 9*0572482 001345 744 553536 41183P784 27*2763634 90613098 001344 745 555025 413493625 27*2946881 9*0653677 001342 746 556516 415160936 27*3130006 9*0694220 001340 T47 558009 416832723 27*3313007 9*0734726 001338 48 559504 418508992 27*3495887 9*0775197 001336 749 561001 420189749 27-3678644 9*0815631 -001335 750 562500 421875000 27*3861279 9*0856030 001333 751 5640011423564751 27-4043792 9*0896352 001331 752 565504 425259008 27*4226184 9-0936719 001329 753 567009 426957777 27*4408455 9-0977010 001328 754 568516 428661064 27*4590604 9-1017265 001326 755 570025 430368875 27*4772633 9-1057485 001324 756 571536 432081216 27*4954542 9*1097669 001322 757 573049 433798093 27*5136330 9-1137818 001321 758 574564 435519512 27*5317998 9-1177931 001319 759 576081 437245479 27*5499546 9-1218010 -001317 760 577600 438976000 27-5680975 9*1258053 001315 761 579121 440711081 27-5862284 9*1298061 001314 762 580644 442450728 27-6043475 9-1338034 001312 763 582169 444194947 27-6224546 9-1377971 001310 764 583696 445943744 27-6405499 9-1417874 001308 765 585225 447697125 27*6586334 9-1457742 001307 766 586756 449455096 27-6767050 9*1497576 001305 767 588289 4512176631 27-6947648 9-1537375 001303 768 589824 452984832 ! 27-7128129 9-1577139 001302 769 591361 454756609' 27-7308492 9-1616869 001300 770 592900 456533000 ; 27*7488739 9-1656565 001298 771 594441 45831401 Ij 27-7668868 9-1696225 001297 772 595984 4P0099648 27-7848880 9-1735852 001295 773 597529 461889917 27-8028775 9-1775445 001293 774 599C76 463684824 27-8208555 9-1815003 001291 775 600625 465484375 27-3388218 9-1854527 00129C 776 602176 467288576 27-8567766 9-1894018 001288 777 603729 469097433 27-8747197 9-1933474 001287 778 605284 470910952 27-8926514 9-1972897 001285 779 606841 472729139 27-9105715 9*2012286 001283 780 60*400 474552000 27*9284801 9*2051641 001282 TABLE OP SQUARES, CUBES, SQUARE ANT> CUBE ROOTS. Number, j Squares. Cubes. ^ Roots. ^Koota. Reciprocals. 781 609061 476379541 27-9463772 9-2090962 00128041 782 611524 478211768 27-9642629 9-2130250 00127877 783 613089 480048687 27-9821372 9-2169505 00127713 784 614656 481890304 28-0000000 9-2208726 00127551 785 616225 i 483736625 28-0178515 9-2247914 00127388 786 617796 485587656 28-0356915 9-2287068 00127226 787 619369 487443403 28-053^203 9-2326189 00127064 788 620944 489303872 28-0713377 9-2365277 00126903 789 622521 491169069 28-0891438 9-2404333 00126742 790 624100 493039000 28-1069386 9-2443355 00126582 791 625681 494913671 28-1247222 9-2482344 00126422 792 627264 496793088 28-1424946 9-2521300 001262B2 793 628849 498677257 28-1602557 9-2560224 001261D3* 794 630436 500566184 28-1780056 9-2599114 00125944 795 632025 502459875 28-1957444 9-2637973 00125786- 796 633616 504358336 28-2134720 9-2676798 OC125628 797 635209 506261573 28-2311884 9-2715592 CG125470 798 636804 508169592 28-2488938 9-2754352 OCI25313, 799 638401 510082399 28-2665881 9-2793081 00125136' 800 640000 512000000 28-2842712 9-2831777 001250001 801 641601 513922401 28-3019434 9-2870444 00124843* 802 643204 515849608 28-3196045 9-2909072 00124688; 803 644809 517781627 28-3372546 9-2947671 001245331 804 646416 519718464 28-3548938 ! 9-2986239 00124378 1 805 648025 521660125 28-3725219 9-3024775 00124223( 806 649636 523606616 28-3901391 9-3063278 00124069* 807 651249 525557943 28-4077454 9-3101750 00123915' 808 652864 527514112 28-4253408 9-3140190 00123762^ 809 654481 529475129 28-4429253 9-3178599 00123609^ 810 656100 531441000 28-4604989 9-3216975 00123456* 811 657721 533411731 28-4780617 9-3255320 00123304* 812 659344 535387328 28-4956137 9-3293634 00123152; 813 660969 537357797 28-5131549 9-3331916 0012'30015 814 662596 539353144 28-5306852 9-3370167 00122850J 815 664225 541343375 28-5482048 9-3408386 /001226994 816 665856 543338496 28-5657137 9-3446575 00122549$ 817 667489 545338513 28-5832119 9-3484731 00122399C 818 669124 547343432 28-6006993 9-3522857 001222494 819 670761 549353259 28-6181760 93560952 001221001 820 672400 5513680GO 28-6356421 0-3599016 001219512 821 674041 553387661 28-6530976 9-3637049 001218027 822 675684 555412248 28-6705424 9-3675051 001216545 823 677329 557441767 28-6879716 9-3713022 007215067 824 678976 559476224 28-7054002 9-3750963 001213592 825 680625 561515625 28-7228132 9-3788873 001212121 826 682276 563559976 28 7402157 9-3826752 001210654 827 683929 565609283 28-7576077 9-3864600 00120919C 828 685584 567663552 28-7749891 9-3902419 00120772S 829 687241 569722789 28-7923601 9-3940206 001206273 830 688900 571787000 28-8097206 9-3977964 001204819 831 690561 573856191 28-8270706 9-4015691 001203369 832 692224 575930368 28-8444102 9-4053387 001201923 TAJBUS OF SQUARES, CUBES, SQUARE AITO CUBE ROOTS. Number. Squares. Cubes. v / Roots. ff Roots. Reciprocals 833 693889 578009537 28-86173.94 9-4091054 001200480 834 695556 5S0093704 28-8790582 9-412SC90 001199041 835 697225 582182875 28-8963666 9-4166297 001197605 836 698896 584277056 28-9136646 9-4203873 001196172 837 700569 586376253 28-9309523 9-4241420 001194743 833 702244 588480472 23-9482297 9-4278936 001193317 839 703921 590589719 28-9654967 9-4316423 001191895 8-10 705SOO 592704000 28-9827535 9-4353800 001190476 841 707281 594823321 29-0000000 9-4391307 001189061 812 708964 5969476S8 29-0172363 9-4428704 001187648 813 710649 599077107 29-0344623 9-4466072 001186240 844 712336 601211584 29-0516781 9-4503410 001184834 845 714025 603351125 29-0688837 9-4540719 001183432 846 715716 605495736 29-0860791 9-4577999 001182033 847 717409 607645423 29-1032644 9-4615249 001180638 848 719104 609300192 29-1204396 9-4652470 001179245 849 720801 611960049 29-1376046 9-4689661 001177856 850 722500 614125000 29-1547595 9-4726824 001176471 851 724201 616295051 29-1719043 9-4763957 001175088 852 725904 618470208 29-1890390 9-4801061 001173709 853 727609 620650477 29-2061637 9-4838136 001172333 854 729316 622835864 29-2232784 9-4875182 001170960 855 856 731025 732736 625026375 627222016 29-2403830 29-2574777 9-4912200 9-4949188 001169591 001168224 857 734449 629422793 29-2745623 9-4986147 001166861 858 736164 631628712 29-2916370 9*5023078 001165501 859 737881 633839779 29-3087018 9-5059980 001164144 860 739600 636056000 29-3257566 9-5096854 001162791 801 741321 638277381 29-3428015 9-5133699 001161440 862 743044 640503928 29-3598365 9-5170515 001160093 863 744769 642735647 29-376S616 9-5207303 001158749 864 746496 644972544 29-3938769 9-5244063 001157407 865 748225 647214625 29-4108323 9-5280794 001156069 866 749956 649461896 29-4278779 9-5317497 001154734 867 751689 651714363 29-4448637 9-5354172 001153403 868 753424 653972032 29-4618397 9-5390818 001152074 869 755161 656234909 29-4788059 9-5427437 001150748 870 756900 658503000 29-4957624 9-5464027 001149425 871 758641 660776311 29-5127091 9-5500589 001148106 872 760384 663054848 29-5296461 9-5537123 001146789 873 762129 1 665338617 29-5465734 9-5573630 001145475 874 763876 667627624 29-5634910 9-5610108 001144165 875 765625 669921875 i 29-5803989 9-5646559 001142857 876 767376 672221376J 29-5972972 9-5682782 001141553 877 769129 674526133 29-6141858 9-5719377 001140251 873 770884 676836152 29-6310648 9-5755745 001138952 879 772641 679151439 29-6479342 9-5792085 001137656 880 774400 681472000 29-6647939 9-5828397 001136364 881 776161 j 683797841 29-6816442 9-5864682 001135074 882 777924 686128968 29-6984848 9-5900937 001133787 883 779689 688465387 29-7153159 9-5937169 001132503 884 781456 1690807104 29-7321375 9-5973373 001131222 TABLE OP SQUARES, CUBES, SQUARE AITD CUBE ROOTS. Number. Squares. Cubes. V Roots. ty Roots. Reciprocals. 885 783225 693154125 29-7489496 9-6009548 00112994'! 886 784996 695506456 29*7657521 9-6045696 001128668 887 - W6T69 697864103J 29-7825452 9-6081817 001127396 S88 788544 700227072 ', 29-7993289 ,.9-6117911 001126126 889 790321 702595369 29-8161030 9-6153977 001124859 89S 792100 i 704969000 : 29-8328678 9-6190017 001123596 891 793881 | 707347971 29-8496231 9-6226030 001122334 892 795664 707932288 29-8663690 9-6262016 001121076 893 797449 712121957 29-8831056 9-6297975 001119821 894 799236 714516984 29-8998328 9-6333907 001118568 895 801025 716917375 29-9165506 9-6369812 001117818 896 802816 719323136 29-9332591 9-6405690 001116071 897 804609 721734273 29-9499583 9-6441542 001114827 898 806404 724150792 29-9666481 9-6477367 001113586 899 808201 726572699 29-9833287 9-6513166 001112347 900 810000 729000000 30-0000000 9-6548933 001111111 901 811801 731432701 30-0166621 9-6584684 001109878 902 813604 733870808 30-0333148 9-6620403 001108647 903 815409 736314327 30-0499584 9-6656096 001107420 904 817216 738763264 30-0665928 9-6691762 001106195 905 819025 741217625 30-0832179 9-6727403 001104972 906 820836 743677416 30-0998339 9-6763017 001103753 907 822649 746142643 30-1164407 9-6798604 001102536 908 824464 748613312 20-1330383 9-6834166 001101322 909 826281 751089429 30-1496269 9-6869701 001100110 910 828100 753571000 30-1662063 9-6905211 001098901 911 829921 756058031 30-1827765 9-6940694 001097695 912 831744 758550825 30-1993377 9-6976151 001096491 913 83356^ 761048497 30-2158899 9-7011583 001095290 914 831)396 763551944 30-2324329 9-7046989 001094092 915 837225 766060875 30-2489669 9-7082369 001092896 916 839056 768575296 30-2654919 9-7117723 001091703 917 840889 771095213 30-2820079 9-7153051 001090513 918 842724 773620632 30-2985148 9-7188354 001089325 919 844561 776151559 30-3150128 9-7223631 001088139 920 846400 778688000 30-3315018 9-7258883 001086957 921 848241 781229961 30-3479818 9.7294109 001085776 922 850084 783777448 30-3644529 9-7329309 001084599 923 851929 786330467 30-3809151 9-7364484 001083423 924 853776 788889024 30-3973683 9-7399634 001082251 925 855625 791453125 30-4138127 9-7434758 001081081 926 857476 794022776 30-4302481 9-7469857 001079914 927 859329 796^97983 30-4466747 9-7504930 001078749 928 861184 799178752 30-4630924 9-7539979 001077586 929 863041 801765089 30-4795013 9-7575002 001076426 930 864900 804357000 30-4959014 9-7610001 001075269 931 866761 806954491 30-5122926 9-7644974 001074114 932 868624 809557568 30-5286750 9-7679922 001072961 933 870489 812166237 30-5450487 9-7714845 -001071811 934 872356 814780504 30-5614136 9-7749743 001070664 935 874225 817400375 30-5777697 9-7784616 001069519 936 876096 820025856 30-5941171 9-7819466 001068376 APPENDIX. Haskins' Differential. In this form of galvano- meter no rheostat is used. Two coils of equal power are placed, one above, and the other below, the needle. The upper coil is raised and lowered by a screw. Attached to the top of the upper coil is a segment of a circle, on which rests a lever with a knife-edge agate terminal to prevent wearing. On the top of the instrument-box is a circular dial, divided into five ohm graduations. A hand, bearing a vernier on its free extremity, is moved by the upper coil through the medium of the segment and the lever above de- scribed. The resistance to be measured is inserted in the circuit of the lower, stationary coil. When the key is closed the upper coil gets more current than the lower, and deflects the needle. This coil is now raised until, owing to its greater distance from the needle, its effect is the same as the lower coil, and the needle returns to zero. The hand on the dial now gives the exact resistance in ohms reading to half an ohm. In this instrument it will be seen that distance, instead of artificial resistance, is used to ob- tain a balance. Galvanometers are made upon this plan with a capacity of 2,000 ohms measurement, but a few inches in size, which may readily be carried MI a coat-pocket. BUNIVBRSITTfl ** Any book in this Catalogue sent free fty tuaS OH rectipt o VALUABLE SCIENTIFIC BOOKS fUBLISHED BY D. VAN NOSTRAND, 23 MURRAY STREET AND 27 WARREN STREET, N. Y. ADAMS (J. W.) Sewers and* Drains for Populous Districts. Embracing- Rules and Formulas lor the dimensions and construction of works of Sanitary Engineers. Second edi- tion. 8vo, cloth $250 ALEXANDER (J. H.) Universal Dietionary of Weights and .Measures, Ancient and Modem, reduced to the standards of the United States of America. New edition, enlarged. 8vo, cloth 350 ATWOOD (GEO.J Practical Blow-Pipe Assaying. i2mo, cloth, illustrated 2 OO AUCH INC LOSS (W. S .). Link and Valve Motions Simplified. 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