Please Note: This item is subject to 
 it CALL after one week. 
 
 _PATI 
 
 "# 
 
 CALIFO^^V
 
 SAE 
 
 3 1822 01155 6347 
 
 CIFT Di <v 
 
 DR. GEORGE F. McEWEN 
 

 
 Columbia Onitoergitp 3lecture 
 
 GRAPHICAL METHODS 
 
 ERNEST KEMPTON ADAMS RESEARCH FUND 
 1909-1910
 
 COLUMBIA 
 
 UNIVERSITY PRESS 
 SALES AGENTS 
 
 NEW YORK : 
 
 LEMCKE & BUECHNER 
 30-32 WEST 2?TH STREET 
 
 LONDON : 
 
 HENRY FROWDE 
 AMES CORNER, E.G. 
 
 TORONTO : 
 
 HEXRY FROWDE 
 
 25 RICHMOND STREET, W.
 
 COLUMBIA UNIVERSITY LECTURES 
 
 GRAPHICAL METHODS 
 
 BY 
 
 CARL RUNGE, PH.D. 
 
 PROFESSOR OF APPLIED MATHEMATICS IN THE UNIVERSITY OP GOTTINGEN 
 
 KAISER WILHELM PROFESSOR OF GERMAN HISTORY AND INSTITUTIONS 
 
 FOB THE TEAR 1909-1910 
 
 JQeto 
 
 COLUMBIA UNIVERSITY PRESS 
 1912
 
 LANCASTER. PA.
 
 INTRODUCTION. 
 
 1. A great many if not all of the problems in mathematics 
 may be so formulated that they consist in finding from given 
 data the values of certain unknown quantities subject to certain 
 conditions. We may distinguish different stages in the solution 
 of a problem. The first stage we might say is the proof that the 
 quantities sought for really exist, that it is possible to satisfy 
 the given conditions or, as the case may be, the proof that it is 
 impossible. In the latter case we have done with the problem. 
 Take for instance the celebrated question of the squaring of the 
 circle. We may in a more generalized form state it thus: Find 
 the integral numbers, which are the coefficients of an algebraic 
 equation, of which TT is one of the roots. Thirty years ago 
 Lindemann showed that integral numbers subject to these con- 
 ditions do not exist and thus a problem as old almost as 
 human history came to an end. Or to give another instance 
 take Fermat's problem, for the solution of which the late Mr. 
 Wolfskehl, of Darmstadt, has left $25,000 in his will. Find the 
 integral numbers x, y, z that satisfy the equation 
 
 where n is an integral number greater than two. Fermat main- 
 tained that it is impossible to satisfy these conditions and he is 
 probably right. But as yet it has not been shown. So the 
 solution of the problem may or may not end in its first stage. 
 In many other cases the first stage of the solution may be so 
 easy, that we immediately pass on to the second stage of finding 
 methods to calculate the unknown quantities sought for. Or 
 even if the first stage of the solution is not so easy, it may be 
 expedient to pass on to the second stage. For if we succeed in 
 finding methods of calculation that determine the unknown quan-
 
 VI GRAPHICAL METHODS. 
 
 titles, the proof of their existence is included. If on the other 
 hand, we do not succeed, then it will be time enough to return 
 to the first stage. 
 
 There are not a small number of men who believe the task of 
 the mathematician to end here. This, I think, is due to the 
 fact that the pure mathematician as a rule is not in the habit of 
 pushing his investigation so far as to find something out about the 
 real things of this world. He leaves that to the astronomer, to 
 the physicist, to the engineer. These men, on the other hand, 
 take the greatest interest in the actual numerical values that 
 are the outcome of the mathematical methods of calculation. 
 They have to carry out the calculation and as soon as they do so, 
 the question arises whether they could not get at the same result 
 in a shorter way, with less trouble. Suppose the mathematician 
 gives them a method of calculation perfectly logical and con- 
 clusive but taking 200 years of incessant numerical work to 
 complete. They would be justified in thinking that this is not 
 much better than no method at all. So there arises a third stage 
 of the solution of a mathematical problem in which the object is 
 to develop methods for finding the result with as little trouble as 
 possible. I maintain that this third stage is just as much a 
 chapter of mathematics as the first two stages and it will not do 
 to leave it to the astronomer, to the physicist, to the engineer or 
 whoever applies mathematical methods, for this reason that 
 these men are bent on the results and therefore they will be apt 
 to overlook the full generality of the methods they happen to 
 hit on, while in the hands of the mathematician the methods 
 would be developed from a higher standpoint and their bearing 
 on other problems in other scientific inquiries would be more 
 likely to receive the proper attention. 
 
 The state of affairs today is such that in a number of cases the 
 methods of the engineer or the surveyor are not known to the 
 astronomer or the physicist, or vice versa, although their prob- 
 lems may' be mathematically almost identical. It is particularly 
 so with graphical methods, that have been invented for definite
 
 INTRODUCTION. Vll 
 
 problems. A more general exposition makes them applicable 
 to a vast number of cases that were originally not thought of. 
 In this course I shall review the graphical methods from a 
 general standpoint, that is, I shall try to formulate and to teach 
 them in their most generalized form so as to facilitate their 
 application in any problem, with which they are mathematically 
 connected. 1 The student is advised to do practical exercises. 
 Nothing but the repeated application of the methods will give 
 him the whole grasp of the subject. For it is not sufficient to 
 understand the underlying ideas, it is also necessary to acquire a 
 certain facility in applying them. You might as well try to learn 
 piano playing only by attending concerts as to learn the 
 graphical methods only through lectures. 
 
 1 For the literature of the subject see " Encyklopadie der mathematischen 
 Wissenschaften," Art. R. Mehmke, " Numerisches Rechnen," and Art. F 
 Willers and C. Runge, "Graphische Integration."
 
 TABLE OF CONTENTS. 
 1. Introduction. . 
 
 CHAPTER I. Graphical Calculation. 
 
 2. Graphical arithmetic 1 
 
 3. Integral functions 6 
 
 4. Linear functions of any number of variables 18 
 
 5. The graphical handling of complex numbers 25 
 
 CHAPTER II. The Graphical Representation of Functions of One 
 or More Independent Variables 
 
 6. Functions of one independent variable 40 
 
 7. The principle of the slide rule 43 
 
 8. Rectangular coordinates with intervals of varying size 52 
 
 9. Functions of two independent variables 58 
 
 10. Depiction of one plane on another plane . . 65 
 
 11. Other methods of representing relations between three 
 
 variables 84 
 
 12. Relations between four variables 94 
 
 CHAPTER III. The Graphical Methods of the Differential and 
 Integral Calculus. 
 
 13. Graphical integration 101 
 
 14. Graphical differentiation 117 
 
 15. Differential equations of the first order 120 
 
 16. Differential equations of the second and higher orders 136
 
 CHAPTER I. 
 GKAPHICAL CALCULATION. 
 
 2. Graphical Arithmetic. Any quantity susceptible of mensu- 
 ration can be graphically represented by a straight line, the 
 length of the line corresponding to the value of the quantity. 
 But this is by no means the only possible way. A quantity 
 might also be and is sometimes graphically represented by an 
 angle or by the length of a curved line or by the area of a square 
 or triangle or any other figure or by the anharmonic ratio of four 
 points in a straight line or in a variety of other ways. The 
 representation by straight lines has some advantages over the 
 others, mainly on account of the facility with which the ele- 
 mentary mathematical operations can be carried out. 
 
 What is the use of representing quantities on paper? It is a 
 convenient way of placing them before our eye, of comparing 
 them, of handling them. If pencil and paper were not as cheap 
 as they are, or if to draw a line were a long and tedious under- 
 taking, or if our eye were not as skillful and expert an assistant, 
 graphical methods would lose much of their significance. Or, 
 on the other hand, if electric currents or any other measurable 
 quantities were as cheaply and conveniently produced in any 
 desired degree and added, subtracted, multiplied and divided 
 with equal facility, it might be profitable to use them for the 
 representation of any other measurable quantities, not so easily 
 produced or handled. 
 
 The addition of two positive quantities represented by straight 
 lines of given length is effected by laying them off in the same 
 direction, one behind the other. The direction gives each line a 
 beginning and an end. The beginning of the second line has to 
 coincide with the end of the first, and the resulting line represent- 
 ing the sum of the two runs from the beginning of the first to 
 2 1
 
 GRAPHICAL METHODS. 
 
 the end of the second. Similarly the subtraction of one positive 
 quantity from another is effected by giving the lines opposite direc- 
 tions and letting the beginning of the line that is to be subtracted 
 coincide with the end of the other. The result of the subtrac- 
 tion is represented by the line that runs from the beginning of 
 the minuend to the end of the subtrahend. The result is positive 
 when this direction coincides with that of the minuend, and nega- 
 tive when it coincides with that of the subtrahend. This leads 
 to the representation of positive and negative quantities by lines of 
 opposite direction. The subtraction of one positive quantity from 
 another may then be looked upon as the addition of a positive and 
 a negative quantity. I do not want to dwell on the logical explana- 
 tion of this subject, but I want to point out the practical method 
 used for adding a large number of positive and negative quantities 
 represented by straight lines of opposite direction. Take a 
 straight edge, say a piece of paper folded over so as to form a 
 straight edge, mark a point on it, and assign one of the two 
 directions as the positive one. Lay the edge in succession over 
 the different lines and run a pointer along it through an amount 
 equal in each case to the length of the line and in the positive 
 or negative direction according to the sign of the quantity. The 
 pointer is to begin at the point marked. The line running from 
 this point to where the pointer stops represents the sum of the 
 
 given quantities. The advan- 
 tage of this method is that the 
 intermediate positions of the 
 pointer need not be marked pro- 
 vided only that the pointer keeps 
 its position during the move- 
 ' ment of the edge from one line 
 to the next. As an example take 
 the area, Fig. 1. A number of 
 
 rectangular strips j cm. wide are substituted for the area so that, 
 measured in square centimeters, it is equal to half the sum of 
 the lengths of the strips measured in centimeters. The straight 
 
 FIG 1.
 
 GRAPHICAL CALCULATION. 3 
 
 edge is placed over the strips in succession and the pointer is 
 run along them. The edge is supposed to carry a centime- 
 ter scale and the pointer is to begin at zero. The final position 
 of the pointer gives half the value of the area in square centi- 
 meters. The drawing of the strips may be dispensed with, their 
 lengths being estimated, only their width must be shown. If 
 the scale should be too short for the whole length, the only thing 
 we have to do is to break any of the lengths that range over the 
 end of the scale and to count how many times we have gone 
 over the whole scale. I have found it convenient to use a little 
 pointer of paper fastened on the runner of a slide rule so that it 
 can be moved up and down the metrical scale on one side of the 
 
 FIG. 2. 
 
 slide rule. The area is in this manner determined rapidly and 
 with considerable accuracy, very well comparable to the ac- 
 curacy of a good planimeter. If the area of any closed curve 
 is to be found, the way to proceed is to choose two parallel 
 lines that cut off two segments on either side (see Fig. 2), to 
 measure the area between them by the method described above 
 and to estimate the two segments separately. If the curves of 
 the segments may with sufficient accuracy be regarded as arcs 
 of parabolas the area would be two thirds the product of length 
 and width. If not they would have to be estimated by substitut- 
 ing a rectangle or a number of rectangles for them.
 
 GEAPHICAL METHODS. 
 
 In the same way the addition and subtraction of pure numbers 
 may also be carried out. We need only represent the numbers 
 by the ratios of the lengths of straight lines to a certain fixed 
 line. The ratio of the length of the sum of the lines to the length 
 of the fixed lines is equal to the sum of the numbers. The con- 
 struction also applies to positive and negative numbers, if we 
 represent them by the ratio of the length of straight lines of 
 opposite directions to the length of a fixed line. 
 
 In order to multiply a given quantity c by a given number, 
 let the number be given as the ratio of the lengths of two straight 
 lines a/6. If the quantity c is also represented by a straight line, 
 all we have to do is to find a straight line x whose length is to 
 the length of c as a to b. This can be done in many ways by 
 
 FIG. 3. 
 
 FIG. 4. 
 
 constructing any triangle with two sides equal to a and 6 and 
 drawing a similar triangle with the side that corresponds to b made 
 equal to c. As a rule it is convenient to draw a and b at right 
 angles and the similar triangle either with its hypotenuse parallel 
 (Fig. 3) or at right angles (Fig. 4) to the hypotenuse of the first 
 triangle. Division by a given number is effected by the same con- 
 struction; for the multiplication by the ratio a/b is equivalent 
 to the divisions by the ratio b/a. 
 
 If a, b, c are any given numbers, we can represent them by the 
 ratios of three straight lines to a fixed line. Then the ratio of
 
 GRAPHICAL CALCULATION. 
 
 the line constructed in the way shown in Fig. 3 and Fig. 4 to 
 the fixed line is equal to the number 
 
 ac 
 
 Multiplication and division are in this way carried out simul- 
 taneously. In order to have multiplication alone, we need only 
 make b equal 1 and in order to have division alone, we need only 
 make a or c equal 1. 
 
 In order to include the multiplication and division of positive 
 and negative numbers we can proceed in the following way. Let 
 the lines corresponding to a, x, Fig. 3, be drawn to the right side 
 of the vertex to signify positive numbers and to the left side to 
 signify negative numbers. Similarly let the lines corresponding 
 to b, c be drawn upward to signify positive numbers and down- 
 ward to signify negative numbers. Then the drawing of a 
 parallel to the hypotenuse of the rectangular triangle a, b through 
 the end of the line corresponding to c will always lead to the 
 number 
 
 whatever the signs of a, b, c may be. 
 
 The same definition will not hold for the construction of Fig. 4. 
 If the positive direction of the line corresponding to a is to the 
 right and the positive direction of the line corresponding to b is 
 upwards then the positive directions of x and c ought to be such 
 that when the right-angled triangle x, c is turned through an 
 angle of 90 to make the positive direction of x coincident 
 with the positive direction of a, the positive direction of c coin- 
 cides with the positive direction of b. If we wish to have the 
 positive direction of x upward, the positive direction of c would 
 have to be to the left, or if we wish to have the positive direction 
 of c to the right, the positive direction of x would have to be 
 downward. If this is adhered to, the construction for division 
 and multiplication will include the signs.
 
 6 
 
 GRAPHICAL METHODS. 
 
 3. Integral Functions. We have shown how to add, subtract, 
 multiply, divide given numbers graphically by representing them 
 as ratios of the lengths of straight lines to the length of a fixed 
 line and finding the result of the operation as the ratio of the 
 length of a certain line to the same fixed line. By repeating 
 these constructions we are now enabled to find the value of any 
 algebraical expression built up by these four operations in any 
 succession and repetition. Let us see for instance how the values 
 of an integral function of x, that is to say, an expression of the form 
 
 may be found by geometrical construction, where a , Oi a n , x 
 
 are any positive or negative 
 numbers. We shall first as- 
 sume that all the numbers are 
 positive, but there is not the 
 least difficulty in extending 
 the method to the more gen- 
 eral case. 
 
 Now let a , ai, 02, a n 
 signify straight lines laid off 
 on a vertical line that we call 
 the y-axis, one after the other 
 as if to find the straight line 
 
 ifc<C 
 
 FIG. 5. 
 
 The lengths of these lines measured in a conveniently chosen 
 unit of length are equal to the numbers designated by the same 
 letters. In Fig. 5 a runs from the point to point C\, ai from 
 Ci to C z , " a n from C n to C+i. 
 
 Let * be the ratio of the lines Ox and 01, Fig. 5, drawn hori- 
 zontally from to the right. The length 01 is chosen of con- 
 venient size independent of the unit of length that measures the 
 lines o , ai, a n . The length Ox is then defined by the value
 
 GRAPHICAL CALCULATION. 
 
 of the ratio x. Through x and 1 draw lines parallel to the z/-axis. 
 Through C n +i draw a line parallel to Ox, that intersects those two 
 parallels in P n and B n . Draw the line B n C n that intersects the 
 parallel through x in P n _i. Then the height of P n _i above C n 
 will be equal to a n x. For if we draw a line through P n _i parallel 
 to Ox intersecting the y-axis in D n , the triangle C n D n P n -i will be 
 similar to C n Cn+iB n and their ratio is equal to x, therefore 
 C n D n = a n x. Consequently the height of Pn_i above C n -i is 
 equal to C n -\D n = a n x + fln-i- Now let us repeat the same 
 operation in letting the point D n take the part of Cn+i. Through 
 D n draw a line parallel to Ox, that intersects the parallels through 
 x and 1 in Pn-i and B n -i> Draw the line B n -iC n -i that intersects 
 the parallel through x in P n -2 
 Then the height of Pn_2 above 
 C n -i will be equal to 
 
 and the height above C n -z will be 
 equal to 
 
 Continue in the same way. Draw 
 P n _ 2 .B n _ 2 parallel to Ox, draw 
 Bn-zCn-z and find the point P n _3- 
 Then the height of Pn_3 above C n _ 2 will be 
 
 (a n x* + an. 
 and the height of Pn-3 above C r , 
 
 FIG. 6. 
 
 Finally a point P is found (see Fig. 6 for n = 4) by the inter- 
 section of BiCi with the parallel to the i/-axis through x, whose 
 height above is equal to 
 
 a n x n an-i 
 
 Let us designate the line xP Q by y, so that
 
 8 GRAPHICAL METHODS. 
 
 y = a n x n + an-iz"- 1 + f a& + a , 
 
 in the sense that y is a vertical line of the same direction and 
 length as the sum of the vertical lines a n x n , a n r-ix n ~ 1 t a\x, OQ. 
 The same construction holds good for values of x greater than 
 1 or negative. The only difference is that the point x is beyond 
 the interval 01 to the right of 1 or to the left of 0. The negative 
 sign of 
 
 a n x, a n x + an_i, a n x^ + a^ix, etc., 
 
 will signify that the direction of the lines is downward. Nor are 
 any alterations necessary in order to include the case that several 
 or all of the lines a , a\, a n are directed downward and corre- 
 spond to negative numbers. They are laid off on the y-axis in 
 the same way as if to find the sum 
 
 flo + fli + 02 + + On, 
 
 (?+! lying above or below C a according to a a being directed 
 upward or downward. The construction can be repeated for a 
 number of values of x. The points P will then represent the 
 curve, whose equation is 
 
 y = a + dix + + a n x n , 
 
 x and y measuring abscissa and ordinates in independent units 
 of length. 
 
 In order to draw the curve for large values of z a modification 
 must be introduced. It will not do to choose 01 small in order 
 to keep x on your drawing board; for then the lines J5.C. will 
 become too short and thus their direction will be badly defined. 
 The way to proceed is to change the variable. Write for instance 
 X = z/10, so that X is ten times as small as x and write 
 
 Then as 
 
 y Oil JQ jQ2 n jQ n
 
 GRAPHICAL CALCULATION. 9 
 
 we find 
 
 y = Ao + AiX + A 2 X 2 + . . . + A n X\ 
 
 Lay off the lines A , A\ t ^4 n in a convenient scale and let 
 Z play the part that x played before. The curve differs in scale 
 from the first curve and the reduction of scale may be different 
 for abscissas and ordinates but may if we choose be made the 
 same so that it is geometrically similar to the first curve reduced 
 to one tenth. It is evident that any other reduction can be 
 effected in the same manner. By increasing the ratio x/X we 
 enhance the value of A n in comparison to the coefficients of lesser 
 index, so that for the figure of the curve drawn in a very small 
 scale all the terms will be insignificant except A n X n . In this 
 case the points C\, Cz, , C n will very nearly coincide with 
 and only C n +i will stand out. 
 
 It is interesting to observe that the best way of calculating an 
 integral function 
 
 for any value of x proceeds on exactly the same lines as the 
 geometrical construction. The coefficient a n is first multiplied 
 with x and a n _i is added Call the result a n _i'. This is again 
 multiplied by x and a n _ 2 is added. Call this result a n -2'. Con- 
 tinuing in this way we finally obtain a value of a ', which is equal 
 to the value of the integral function for the value of x considered. 
 Using a slide rule all the multiplications with x can be effected 
 with a single setting of the instrument. The coefficients a a and 
 the values a* are best written in rows in this way 
 
 O n drtr-l On-2 " ' Ci\ O,Q 
 
 a n x a n -i'x az'x ai'x 
 
 The accuracy of the slide rule is very nearly the same as the 
 accuracy of a good drawing. But the rapidity is very much 
 greater. When therefore only a few values of the integral func- 
 tion are required, the geometrical construction will not repay
 
 10 
 
 GRAPHICAL METHODS. 
 
 the trouble. It is different, however, when the object is to make 
 a drawing of the curve. The values supplied by calculation 
 would have to be plotted, while the geometrical construction 
 furnishes the points of the curve right away and in this manner 
 gains on the numerical method. 
 
 There is another geometrical method, which in some cases 
 may be just as good. Let us propose to find the value of an 
 integral function of the fourth degree. 
 
 y = a + aix + ctzx 2 + asz 3 + 0,43* 
 
 and let all coefficients in the first instance be positive. 
 The coefficients ao, a-i, 02, as, a 4 are supposed to be represented 
 
 by straight lines, while x will be the ratio of two lines. The lines 
 
 OD, 01, 02, 03, o 4 are laid off in a 
 broken line ao to the right from 
 Co to Ci, ai upward from Ci to 
 C 2 , 02 to the left from C 2 to C 3 , 03 
 downward from C 3 to C 4 , o 4 again 
 to the right from C 4 to C 5 (Fig. 7). 
 Through C 5 draw a line C$A to 
 a point A on C 3 C 4 or its prolonga- 
 tion and let x be equal to the 
 ratio CiA : C 4 C 6 taken positive 
 when CA has the same direc- 
 
 - ^- 
 
 D 
 
 ^ 
 
 J*** as 
 
 > 
 
 \ 
 
 \ 
 
 "\, / 
 
 % 
 
 !fc\ 
 
 \ 
 
 
 \ 
 
 jr ^ / 
 
 \ j 
 
 C a o ' (a 
 
 Fio. 7. 
 
 tion as Cad. Then we have 
 
 and 
 
 i = o 4 z, 
 C 3 A = 0^ + 
 
 (7^4 and Cs^l are positive or negative according to their direction, 
 being the same as the direction of C 3 (7 4 or opposite to it. Through 
 A draw the line AB forming a right angle with C&A to a point B 
 on C z Cz or its prolongation. Then we have 
 
 and 
 
 C 3 A 
 
 o 3 ) x
 
 GRAPHICAL CALCULATION. 
 
 11 
 
 CzB = aix* + azx -f- 02. 
 
 C S B and C 2 B are positive or negative according to their direction 
 being the same as the direction of C 2 Cs or opposite to it. 'Simi- 
 larly we get 
 
 CiD = a^ + asz 2 + 022 + ai, 
 and finally 
 
 a z y? + c^z 2 + fliz + OQ. 
 
 CoE is positive, when E is on the right side of Co and negative 
 
 when on the left side. When the point A moves along the line 
 
 CzCi, the point E will move 
 
 along the line Cod and its 
 
 position will determine the 
 
 values of the integral function. 
 
 To find the position of E for 
 
 any position of A, we might 
 
 use transparent squared paper, /u^ 
 
 that we pin onto the drawing 
 
 at C 5 , so that it can freely be 
 
 turned round 5. Following 
 
 FIG. 8. 
 the lines of the squared paper 
 
 along C&ABDE after turning it through a small angle furnishes 
 the position of E for a new position of A (Fig. 8). 
 
 To include the case of negative coefficients we draw the corre- 
 sponding line in the opposite direction. If for instance 03 is 
 negative C 3 C 4 would have to lie above C 3 ; but C 3 A would have 
 to be counted in the same way as before, positive in a downward, 
 negative in an upward direction. 
 
 The extension of the method to integral functions of any degree 
 is obvious and need not be insisted on. It may be applied with 
 advantage to find the real roots of an equation of any degree. 
 For this purpose the broken line C^ABDE would have to be 
 drawn in such a way that E coincides with CQ. In the case of 
 Fig. 7, for instance, it is easily seen that no real root exists. 
 Fig. 9 shows the application to the quadratic equation. A circle
 
 12 
 
 GRAPHICAL METHODS. 
 
 is drawn over CoC 3 as diameter. Its intersections with 
 furnish the points A and A' that correspond to the two roots. 
 Both roots are negative in this case. 
 
 The first method of constructing 
 the values of an integral function can 
 be extended to the case where the 
 function is given as the sum of a 
 number of polynomials of the form 
 
 y = a Q + ai(x p) + a*(x p) (x q) 
 + a s (x - p)(x - q)(x - r) + . 
 
 Let us again suppose a , ai, 02, 
 to represent straight lines laid off as 
 before on the ?/-axis upwards or down- 
 
 wards as if to find their sum. x,p,q,r are meant, to be num- 
 bers represented by the ratio of certain segments on the axis of 
 abscissas. Let us consider the case of four terms, the highest poly- 
 nomial being of the third degree. The fixed distance between the 
 points marked p and p + 1, q and q-\- I, r and r -f- 1 on the 
 axis of abscissas, Fig. 10 is chosen arbitrarily and the position 
 
 ^ 
 
 r 
 
 
 
 \ 
 
 
 ^ 
 
 Qo 
 
 " 
 
 
 
 
 
 ,'"'' 
 
 
 
 1 
 
 -G 
 
 
 A- 
 
 -::i^ 
 
 "Q> 
 
 1 
 
 
 "'( 
 
 2 
 
 
 
 \ 
 
 
 
 
 
 p 2 r x JM-1 gfl <rtl o;^ 
 FIG. 10. 
 
 of the points marked p, q, r, x is made such that the ratio of 
 Op, Oq, Or, Ox to that fixed distance is equal to the numbers 
 p, q, r, x. For negative values the points are taken on the left 
 of 0.
 
 GRAPHICAL CALCULATION. 13 
 
 Draw parallels to the ?/-axis through p, q, r, x, p + 1, q + 1, 
 r + 1. On the parallel through r + 1 find the point Qo of the 
 same ordinate as C\ and on the parallel through r find the point 
 AQ of the same ordinate as C& Join AQ and QQ by a straight 
 line and find its intersection PI or that of its prolongation with 
 the parallel through x. The height of PI above C 3 or AQ is 
 equal to 03(2 r) and the height above Cz is equal to a 3 (x r) 
 + 02. On the parallel through q + 1 find a point Qi of the same 
 ordinate as PI and on the parallel through q a point A\ of the 
 same ordinate as Cz. Join A\ and Qi by a straight line and find 
 its intersection P 2 or that of its prolongation with the parallel 
 through x. The height of PZ above Cz or A\ is equal to 
 
 [a z (x r) + av](x q), 
 and the height above C\ is equal to 
 
 a 3 (x - r)(x - q) + a*(x - q) + i. 
 
 Finally find a point Qz on the parallel through p + 1 of the 
 same ordinate as P 2 and a point A 2 on the parallel through p of 
 the same ordinate as C\. Join ^4 2 and Qz by a straight line and 
 find its intersection P$ or that of its prolongation with the par- 
 allel through x. The height of P 3 above Ci or A z will then be 
 equal to 
 
 MX - r)(x - q) + az(x - q) + a^(x - p) 
 
 and the ordinate of P 3 will be equal to the given integral function 
 
 y = a 3 (x - r)(x - q)(x - p) + a*(x - q)(x - p) 
 
 + ai(z p) + a . 
 
 For large numbers p, q, r, x we use a similar device as before by 
 introducing new numbers P, Q, R, X equal to one tenth, or one 
 hundredth or any other fraction of pqrx. For instance 
 
 P = p/10, Q = 9/10, R = r/10 X = r/10. 
 We then write 
 
 A Q = a , Ai = 10a b A 2 = 10002, A 3 = 1000a 3 ,
 
 14 GRAPHICAL METHODS. 
 
 and obtain 
 
 y = A, + A 1 (X -P) + A 2 (X - P)(X - Q) 
 
 + A*(X - P)(X - Q)(X - R). 
 
 The scale for the lines A , A\, A z , A 3 and y must then be reduced 
 conveniently and the values are constructed in the same way as 
 before. 
 
 Now let us consider the inverse problem. The values of the 
 integral function are given for 
 
 x = p, q, r, s; 
 
 find the lines a , a\ t (h, OB, so that the value of the integral function 
 may be found for any other value of x in the way shown above. 
 
 Let us designate the given values of the integral function for 
 x = p, q, r, s by y p , y q , y r , y 9 and the points on the parallels through 
 p, q, r, s with these ordinates by P, Q, R, S (see Fig. 12). 
 
 For x = p the integral function 
 
 y = a + ai(x p) + 02(2 p)(x q} + a z (x p)(xq}(xr} 
 
 reduces to a . Therefore we have y p = a . The point Ci is 
 found by drawing a parallel to the axis of abscissas through P 
 
 and taking its intersection with 
 the axis of ordinates. 
 ^-T|A In order to find C 2 draw a 
 
 straight line through P and Q 
 
 and find its intersection A with 
 ~\B the parallel through p + 1 (Fig. 
 
 11). A parallel to the axis of 
 abscissas through A intersects 
 
 $ the axis of ordinates in C 2 . For 
 
 FlG llm the differences y q y p and y a y p 
 
 (writing y a for the ordinate of 
 
 A) are proportional to the differences of the abscissas and con- 
 sequently in the ratio (q p) : 1. Therefore 
 
 *--**-*
 
 GRAPHICAL CALCULATION. 
 
 15 
 
 In the same way as the point Q on the parallel through q we 
 might join any point X on a parallel through x with the point P, 
 find the intersection with the parallel through p + 1 and draw a 
 parallel to the axis of abscissas. The point of intersection of 
 
 FIG. 12. 
 
 this parallel with the vertical through x let us call X' and its 
 ordinate y'. Then we have 
 
 y - yp = 
 
 - q) + a s (x - q) (x r). 
 
 y- y P ^ a , 
 
 x- p 
 
 Let us carry out this construction not only for x = q but also 
 for x = r and x = s. This leads us to three points Q', R f , S' 
 on the verticals through q, r, s, whose ordinates are the values 
 of the integral functions 
 
 y' = (a + 
 
 q)(x r). 
 
 In this way we have reduced our problem. Instead of having 
 to find an integral function of the third degree from four given 
 points P, Q, R, S, we have now only to find an integral function 
 of the second degree from three given points Q', R', S'. A second 
 reduction is effected in exactly the same manner. Q' is joined 
 with R' and S' by straight lines and through their intersection 
 with the vertical through q + 1 parallels to the axis of abscissas 
 are drawn that intersect the verticals through r and s in the 
 points R" and S" respectively. The ordinates of these points 
 are the values of the integral function y" defined by
 
 16 GRAPHICAL METHODS. 
 
 for x = r and x = s, or 
 
 y" = ao + a\ + 02 + a 3 (z ") 
 
 The horizontal through R" intersects the axis of ordinates in the 
 point C 3 . Finally we find C 4 by drawing a parallel to the axis 
 of abscissas through the intersection of R"S" or its prolongation 
 with the vertical through r + 1. 
 
 Having found the points CiCzCsd we can now for any value 
 of x construct the ordinate 
 
 y = a + ai(x p) + 02(2 p)(x q) 
 
 + a 3 (x- p)(x- q)(x- r), 
 
 and thus draw the parabola of the third degree passing through 
 the four points P, Q, R, S. 
 
 The construction may be somewhat simplified first by making 
 p + 1 = q. Our data are the points P, Q, R, S, and we are 
 perfectly at liberty to make the vertical through p + 1 coincide 
 with the vertical through Q. In this case the point Q' will 
 coincide with Q. The parabola of the second degree through the 
 points Q'R'S' is again independent of the distance between the 
 verticals through q and 9+1 and at the same time independent 
 of the point P. Therefore we are perfectly at liberty, for the 
 construction of any point of this parabola, to make the vertical 
 through q + 1 coincide with the vertical through R even if the 
 distance of the verticals through P and Q is different from that 
 of the verticals through Q and R. R" will in this case coincide 
 with R'. The procedure is shown in Fig. 12. Starting from 
 the points P, Q, R, S the first step is to find R', S' by connecting 
 R and S with P and drawing horizontals through the inter- 
 sections A r , A, with the vertical through q. The next step is to 
 find S" by connecting Q (identical with Q') with S' and drawing 
 a horizontal through the intersection with the vertical through r. 
 Now the straight line R"S" can be drawn (R" being identical
 
 GRAPHICAL CALCULATION, 
 
 17 
 
 with R'). On the vertical through any point x take the inter- 
 section with R"S" and pass horizontally to the point A x ' on the 
 vertical through r. Draw the line Q'A X ' and find its intersection 
 with the vertical through x. This point is on the parabola 
 through Q'R'S'. Pass horizontally to the point A x on the 
 vertical through q and draw the line A X P. Its intersection with 
 the vertical through # is a point on the parabola of the third 
 degree through P, Q, R, S. 
 
 The method is evidently applicable to any number of given 
 points, the degree of the parabola being one unit less than the 
 number of points. 
 
 The methods for the construction of the values of an integral 
 function may be applied to find the value of any rational function 
 
 V = R(x). 
 
 For a rational function can always be reduced to the form of a 
 quotient of two integral functions 
 
 R(x) = g 
 
 Xlf 
 
 Now after having constructed curves whose ordinates give the 
 
 values of g\(x) and gz(x) for any abscissa x (Fig. 13), R(x) is found 
 
 in the following manner. 
 
 Through a point P on the 
 
 axis of abscissas draw a 
 
 parallel to the axis of or- 
 
 dinates. Let GI and G 2 
 
 be the points whose ordi- 
 
 nates are equal to gi(x) 
 
 and gz(x). Pass horizon- 
 
 tally from GI to GI on the 
 
 vertical through P and 
 
 from Gz to Gz on the axis of ordinates. Draw a line through 
 
 P and Gz and produce it as far as A where it intersects the 
 
 horizontal through GI. Then R(x) is equal to the ratio G\A 
 
 to PO. Gi'A may then be set off as ordinate on the vertical 
 
 Fio. 13,
 
 18 
 
 GRAPHICAL METHODS. 
 
 through x and defines the point M whose ordinate is equal to 
 R(x} in length, when OP is chosen as the unit of length. 
 
 4. Linear Functions of Any Number of Variables. Let us 
 consider a linear function of a number of variables xi, x% x n , 
 
 a Q + 
 
 04X2 + 
 
 a n x n , 
 
 where a< ai, a%> o, n are given numbers positive or negative. 
 The question is how the value of this linear function may be 
 conveniently constructed for various systems x\, Xz, x n . 
 Suppose ao, a\, a n to represent horizontal lines directed to 
 the right or left according to the sign of the corresponding number 
 and to be laid off on an horizontal axis in succession as if to find 
 the sum 
 
 ao + ai + 03 + + a n , 
 
 a Q begins at and runs to C\, Oj begins at C\ and runs to (7 2 and 
 so on (Fig. 14). The numbers xi, x^, x n let us represent 
 
 a 8 
 
 FIG. 14. 
 
 by ratios of lengths. We draw a vertical line through and 
 choose a point P on the horizontal axis. Then let xi be equal 
 to the ratio OlfPO, x^ = 02/PO, etc. If P is chosen on the left 
 of 0, we take the point 1 above for a positive value of xi and 
 below for a negative one and the same for the other points. 
 Mark a point above in the same distance from as P. Join 
 the point P with the points 0, 1, 2, 3, 4, and draw a broken
 
 GRAPHICAL CALCULATION. 
 
 19 
 
 line OAA\A<iAzA\ in such a manner that AQ is on the vertical 
 through Ci and OA Q is parallel to PO, A\ on the vertical through 
 <7 2 and A Ai parallel to PI, A 2 on the vertical through C s and A\A^ 
 parallel to P2 and so on. Then the ordinate y of A will have 
 the same length as a and will be directed upward when the 
 direction of o is to the right, and downward when the direction 
 of do is to the left. The difference y\ yo of the ordinates of A\ 
 and AQ is equal in length to CL\XI, as y\ yo and ai have the same 
 ratio as 01 and PO. AI will be above or below AQ according to 
 the line a\x\ being directed to the right or to the left and it is 
 understood that a\x\ has the same direction as a\ for positive 
 
 FIG. 15. 
 
 values of x\ and a direction opposite to ai for negative values 
 of a?i. Thus the ordinate y\ has the same length as the line 
 ffo 4~ d\x\ and its direction is upward or downward according to 
 the direction of the line o + i^i being to the right or to the left. 
 In the same way it is shown that the ordinate y z of the point A 2 
 is equal in length to 
 
 do + CLiXi + 02^2, 
 
 and 2/3 to 
 
 flfl + Ol^l + 02^2 + 3^3 
 
 and so on, the direction upward or downward corresponding 
 to the positive or negative value of the linear function. 
 If the values of x\, x%, x n satisfy the equation 
 
 a + a&i + 02^2 + + d n x n = 
 the ordinate y n must vanish, that is to say, the point A n must
 
 20 
 
 GRAPHICAL METHODS. 
 
 coincide with C n +i, the end of the line a n . And vice versa if A n 
 and Cn+i coincide the equation is satisfied. Consequently if we 
 know all the values but one of the numbers Xi, Q, x n the 
 unknown value can be found graphically. For suppose x 3 to be 
 
 PIG. 16. 
 
 the unknown value we can, beginning from 0, find the broken 
 line as far as A 2 and beginning from the other end A n we 
 can find it as far as A$ (Fig. 15). A parallel to A Z A 3 through P 
 furnishes the point 3 on the axis of ordinates. If Xi, Xz, Zn-i 
 are known and only x n not, we can draw the broken line as far 
 as An-i and as A n has to coincide with C n+ i, we can draw a parallel 
 to An-iA n through P and find the point n on the axis of ordinates 
 
 FIG. 17. 
 
 that determines the value x n by the ratio On/PO or OnfOo. In 
 Figs. 15 and 16 all the coefficients ao, ai, , are positive. A 
 negative coefficient 05 is shown in Fig. 17. The only difference 
 is that Ce lies to the left of C& and consequently the broken line 
 passes from AI back to A& 
 
 If we keep the points 0, 1, 2, , in their positions but change 
 the position of P to P' (Fig. 18) and repeat the construction of
 
 GRAPHICAL CALCULATION. 
 
 21 
 
 the broken line, we obtain OAJAiAj instead of OAoAiA z 
 The ordinate y* of the point A a r is evidently 
 
 00 . 01 Oa 
 
 and therefore 
 
 PO 
 
 That is to say, by changing the position of P without changing 
 the position of the points 0, 1, 2, we can change the scale of 
 the ordinates of the broken line. They change inversely pro- 
 
 
 o (7i 
 
 FIG. 18. 
 
 portional to PO. It may be convenient to make use of this 
 device in order to make the ordinates a convenient size inde- 
 pendent of the scale that we have chosen for the points 0, 1, 2, 
 that determine the values 
 
 _ 01 _02 
 
 xi - Q , a* - , 
 
 A linear equation with only one unknown quantity 
 GO + a&i = 
 
 is solved by drawing a parallel to A Ai through P. Let a second 
 equation be given with two unknown quantities 
 
 xz = 0. 
 
 The lines bo, bi, 62 are laid off as before. Knowing xi as the 
 solution of the first equation we can construct the broken line 
 corresponding to the second equation and as B 2 must
 
 22 GRAPHICAL METHODS. 
 
 coincide with the end of bz, we can draw a parallel to 
 through P and find Xz. In a similar manner we can find z 3 
 from a third equation 
 
 Co + CiZi + CzXz + C S X 3 = 0, 
 
 and so we can find any number of unknown quantities, if 
 each equation contains one unknown quantity more than those 
 before. 
 
 In the general case when n unknown quantities are to be 
 determined from n linear equations each equation will contain 
 all the unknown quantities, and therefore we cannot find them 
 one after the other as in the case just treated. But it can be 
 shown that by means of very simple constructions the general case 
 is reduced to a set of equations, such as has just been treated. 
 
 A A A -Let us begin with two 
 
 equations and two un- 
 
 & 
 
 V'V 
 
 known quantities. 
 
 FlG - 19 - 
 
 The lines a Q , a\, az are laid off on a horizontal line OA AiA z and 
 the lines 6 , 61, 62 on another horizontal line 0'B BiB 2 (Fig. 19). 
 Now let us join and 0', A and B , A\ and BI, A 2 and B 2 by 
 straight lines and let us draw a third horizontal line intersecting 
 them in the points 0"C CiC 2 . These points correspond to a 
 certain linear function 
 
 Co + CiZi + CzZij, 
 
 and it can be shown that it vanishes when xi and Xz are the same 
 values for which the first two linear functions vanish. Let the 
 distance of the first two horizontal lines be I and the distance of 
 the third from the first and second h and k. Then it can readily 
 be seen that 
 
 Co = a + - (6 - o ) = a Q + - 6 .
 
 GRAPHICAL CALCULATION. 23 
 
 For a parallel to 00' through AQ defines with the line A Bo on 
 the third and second horizontal line segments equal to Co flo 
 and 6 OQ and as these segments have the ratio h/l, it follows 
 that 
 
 , h k h 
 
 CQ = a Q + y (0 o ) = y OD + y DO- 
 
 By drawing a parallel to A B Q through A\ and to A\B\ through 
 Az or through 5 2 (which comes to the same thing), we convince 
 ourselves in the same way that 
 
 Ci = oi + y (&i 01) = y ai + y k 
 and 
 
 02 = 02+ y $2 02) = y 02 + y 62. 
 
 Multiplying the equation 
 
 a + OiZi + 02^2 = 
 by fc/7 and the equation 
 
 bo + to + ban = 
 by A// and adding the two products, we obtain 
 
 C + Wi + 2X2 = 0. 
 
 The third horizontal need not lie between the first two. If it 
 lies below the second we have merely to give k a negative value 
 and if it lies above the first we have to give h a negative value 
 and the same formulae for <? , Ci, GZ hold good. Consequently the 
 conclusion remains valid, that from the first two equations the 
 third follows. 
 
 Now as we are perfectly at liberty to draw the third horizontal 
 line where we please, we can let it run through the intersection 
 of the straight lines A\B\ and A^Bz. In this case the points C\ 
 and Cz must coincide and consequently c^ must vanish. If c\ 
 does not vanish we can by what has been shown above find .TI 
 and with x\ we can find a^ from either of the two first horizontal
 
 24 GKAPHICAL METHODS. 
 
 lines. In case c\ also vanishes, that is to say, in case the three 
 straight lines A^Bz, A\B\, A^Bo all pass through the same point, 
 while 00' does not pass through it, the two given equations 
 cannot simultaneously be satisfied. For if they were, it would 
 follow that 
 
 C + CiXi + CzXz = 0, 
 
 and as c\ and 02 are zero c would have to be zero, which it is not 
 as 00' is supposed not to pass through the intersection of A Z B Z , 
 A\B\ and A Q Bo. If on the other hand all four lines A 2 Bz, A\B\, 
 AoBo, 00' pass through the same point, CQ, c\ and c% will all three 
 vanish. In this case the two given equations do not contradict 
 one another, but &o&i&2 will be proportional to aoaiG^. The 
 
 Ai AI A a A t A, A t 
 
 second equation will therefore contain the same relation between 
 xi and xz as the first, so that there is only one condition for Xi 
 and xz to be satisfied. We may then assign any arbitrary value 
 to one of them and determine the value of the other to satisfy the 
 equation. 
 
 In the case of two linear equations of any number of quantities 
 %i) Xz> x n we can by the same graphical method eliminate one 
 of the quantities. In Fig. 20 this is shown for two linear equa- 
 tions with six unknown quantities. The two horizontal lines 
 OAoAiAzAsAiAsAe and 0'BQBiB 2 B 3 B^B 5 B 6 represent two linear 
 equations. Through the intersection of A 3 B 3 and A 4 B 4 a third 
 horizontal line is drawn intersecting the lines 00', AoB , A\B\, 
 - ' - AfiBs in 0"C Q Ci C 6 . As C 3 and (7 4 coincide, the line c 4 
 vanishes and a- 4 is eliminated, so that the equation assumes the 
 form 
 
 Co + G\XI + fyXz+CsXt -f- c&a*5 + CeZe = 0-
 
 GRAPHICAL CALCULATION. 25 
 
 Suppose now that a set of six equations with six unknown quan- 
 tities is represented geometrically on six horizontal lines. We shall 
 keep one of these; but instead of the other five we construct five 
 new ones from which one of the unknown quantities has been 
 eliminated by means of the first equation. Now it may happen 
 that at the same time another unknown quantity is eliminated, 
 then this quantity remains arbitrary. Of the five new equations 
 we again keep one that contains another unknown quantity and 
 replace the four others again by four new ones from which this 
 unknown quantity has been eliminated. Going on in this 
 manner the general rule will be that with each step only one 
 quantity is eliminated, so that at last one equation with one un- 
 known quantity remains. Instead of the given six equations 
 with six unknown quantities each, we now have one with six, 
 one with five and so on down to one with one. The geometrical 
 construction shows that this system is equivalent to the given 
 system, for we can just as well pass back again to the given 
 system. We have seen above how the unknown quantities 
 may now be found geometrically. It may however happen in 
 special cases that with the elimination of one unknown quantity 
 another is eliminated at the same time. To this we may then 
 assign an arbitrary value without interfering with the possibility 
 of the solution. Finally all unknown quantities may be elimi- 
 nated from an equation. If in this case there remains a term 
 different from zero it shows that it is impossible to satisfy the 
 given equations simultaneously. If no term remains, the two 
 equations from which the elimination takes its origin contain the 
 same relation between the unknown quantities and one of them 
 may be ignored. 
 
 5. The Graphical Handling of Complex Numbers. A. complex 
 number 
 
 z = x + yi 
 
 is represented graphically by a point Z whose rectangular coordi- 
 nates correspond to the numbers x and y. The units by which
 
 26 
 
 GRAPHICAL METHODS. 
 
 the coordinates are measured, we assume to be of equal length. 
 We might also say that a complex number is nothing but an 
 algebraical form of writing down the coordinates of a point in a 
 plane. And the calculations with complex numbers stand for 
 certain geometrical operations with the points which correspond 
 to them. 
 
 By the "sum" of two complex numbers 
 
 21 = xi + yii and 22 = a& + y*i 
 we understand the complex number 
 
 where 
 
 and we write 
 
 23 
 
 and y 3 
 
 = 2i + 22. 
 
 Graphically we obtain the point Zz representing 23 from the 
 points Zi and Z 2 representing zi and 22 by drawing a parallel 
 to OZ Z through Z\ and making Z\P 
 (Fig. 21) equal to OZ 2 in length 
 and direction or by drawing a paral- 
 lel through Z 2 and making Z 2 P 
 equal to OZi in length and direc- 
 tion. The coordinates of P are 
 evidently equal to #1 + Xz and 
 2/i + 2/2. 
 
 Two complex numbers 2 and z' 
 are called opposite, when their sum 
 is zero. 
 
 FIG. 21. 
 
 z + z' = or x = x' and 
 
 or 2 = z 
 
 y- - y 
 
 The corresponding points Z and Z' are at the same distance from 
 the origin but in opposite directions. 
 
 The difference of two complex numbers is that complex 
 number, which added to the subtrahend gives the minuend 
 
 22 + (2l %) = 2l-
 
 GRAPHICAL CALCULATION. 
 
 27 
 
 Therefore 
 
 21 % = (zi a*) + (1/1 
 
 This may also be written 
 
 zi + 22' where 22'= 22 = a^ 2/2*. 
 
 That is to say, the subtraction of the complex number 2 2 from zi 
 may be effected by adding the opposite number 22. For the 
 geometrical construction of the point Z corresponding to z\ & 
 we have to draw a parallel to OZ 2 through Z\ and from Z\ in 
 the direction from Z 2 to we have to lay off the distance Z Z 0. 
 Or we may also draw from a line equal in direction and in length 
 to ZzZi. This will also lead to the point Z representing the 
 difference 21 22. 
 
 The rules for multiplication and division of complex numbers 
 are best stated by introducing polar coordinates. Let r be the 
 positive number measuring the distance OZ in the same unit 
 of length in which x and y measure the abscissa and ordinate, so 
 that 
 
 and let <p be the angle between OZ and the axis of x, counted in 
 the direction from the positive axis of x toward the positive 
 axis of y through the entire 
 circumference (Fig. 22). Then 
 we have 
 
 x = r cos <p, y = r sin <p 
 and 
 2 = x + yi = r(cos<f> + sin (pi). 
 
 Let us call r the modulus 
 and tp the angle of z. The an- 
 gle may be increased or di- 
 
 minished by any multiple of four right angles without altering 
 z, but any alteration of r necessarily implies an alteration of z. 
 
 2 2.
 
 28 GRAPHICAL METHODS. 
 
 According to Moivre's theorem, we can write 
 
 z = re**'. 
 By the product of two complex numbers 
 
 zi = rie*" and 22 = tfce*** 
 
 we understand that complex number 23 whose modulus r$ is 
 equal to the product of the moduli r\ and r 2 and whose angle ^ 
 is the sum of the angles <p\ and <& or differs from the sum only by 
 a multiple of four right angles 
 
 The definition of division follows from that of multiplica- 
 tion. The quotient Zi divided by 22 is that complex number, 
 which multiplied by 22 gives Zi. Therefore the product of its 
 modulus with the modulus of Za must be equal to the modulus of 
 Zi and the sum of its angle with the angle of 22 must be equal to 
 the angle of z\. Or we may also say the modulus of the quotient 
 Zi/Z2 is equal to the quotient of the moduli TI/TZ and its angle is 
 equal to the difference of the angles <p\ <& An addition or 
 subtraction of a multiple of four right angles we shall leave out 
 of consideration as it does not affect the complex number nor 
 the point representing it. 
 
 The geometrical construction corresponding to the multi- 
 plication and division of complex numbers is best described by 
 considering two quotients each of two complex numbers that 
 give the same result. Let us write 
 
 2l/Z2 = Z 3 /Z4. 
 
 The geometrical meaning of this is that 
 
 rifa = r 3 /r 4 , 
 and 
 
 <f>i <f>z <f>3 <p\. 
 
 That is to say, the triangles Z^OZz and Z 3 OZ 4 are geometrically
 
 GRAPHICAL CALCULATION. 
 
 29 
 
 similar (Fig. 23). When three of the points Z lt Z 2 , Z 3 , Z 4 are 
 given the fourth can evidently be found. For instance let 
 Z\, Zz, Z\ be given. Draw a parallel to Z\Z^. intersecting OZ% 
 at a distance r 4 from 0. This point together with the inter- 
 section on OZi and with will form the three corners of a tri- 
 angle congruent to the triangle Z 4 Z 3 0. It will be brought into 
 
 FIG. 23. 
 
 FIG. 24. 
 
 the position of Z 4 Z 3 by being turned round so as to bring the 
 direction of the side in OZ 2 into the position of 0Z 4 . Thus the 
 direction of OZ 3 and its length may be found. 
 
 This construction contains multiplication as well as division as 
 special cases. Let Z 4 coincide with the point x = 1, y = 0, so 
 that z 4 = 1 (Fig. 24), then we have 
 
 Zl/Z2 = 3j OF Zi = 22Z3. 
 
 From any two of the points Z\, Z 2 , Z 3 a simple construction gives 
 us the third. 
 
 The geometrical representation of complex numbers may be used 
 to advantage to show the properties of harmonic oscillations. 
 
 Let a point P move on the axis of x, so that its abscissa at the 
 time t is given by the formula 
 
 x = r cos (nt + a), 
 n, r and a being constants. We call r the amplitude and nt + a
 
 30 GRAPHICAL METHODS. 
 
 the phase of the motion. The point P moves backwards and 
 forwards between the limits x = r and x = r. The time 
 T = 2ir/n is called the period of the oscillation, it is the time in 
 which one complete oscillation backwards and forwards is per- 
 formed. 
 
 Now instead of x let us consider the complex number 
 
 z = r cos(nt + a) + r sin(nt + a)i 
 or 
 
 2 = re (n * +a) % 
 
 of which x is the abscissa and let us follow the movement of 
 the point Z. For t = we have 
 
 z = re ai . 
 Designating this value by z , we can write 
 
 z = zoe ntt . 
 The geometrical meaning of the product 
 
 is that the line OZ is turned round through the angle nt. For 
 the modulus of e nti being equal to 1 the modulus of z is not 
 changed by the multiplication. The 
 movement of the point Z therefore 
 consists in a uniform revolution of 
 Z round 0. At the moment t= 
 the position is OZo and after the 
 time T = 2ir/n the same position is 
 
 
 *=2/ 2 ' 
 
 \ J 
 
 occupied again. The revolution goes 
 on in the direction from the positive 
 axis of x to the positive axis of y 
 (Fig. 25). 
 
 FlQ ^ The movement of Z is evidently 
 
 simpler than the movement of the 
 projection P of Z on the axis of x. 
 
 Let us consider a motion composed of the sum of two harmonic
 
 GRAPHICAL CALCULATION. 31 
 
 motions of the same period but of different amplitudes and 
 phases 
 
 x = r i cos (nt + en) + r 2 cos (nt + <* 2 ), 
 
 and let us again substitute the motion of the point Z correspond- 
 ing to the complex number 
 
 2 = 
 
 For t = the first term is 
 and the second term 
 
 Introducing z\ and % into the expression for z we have 
 
 z = Zl e nti + sfee"*' = (21 + ze nti = z 3 e n 
 where 
 
 23 = 2i + 2z. 
 
 This shows at once that the movement of Z is a uniform circular 
 movement consisting in a uniform revolution of OZ round 0. 
 The position at the moment t = is OZ$ corresponding to the 
 complex number 
 
 The projection of Z on the axis of x has the abscissa 
 
 x = TZ cos (nt + 3 ) 
 
 where r 3 and a 3 designate modulus and angle of z 3 . Thus the 
 sum of two harmonic motions of the same period is shown also 
 to form a harmonic motion. 
 
 The same holds for a sum of any number of harmonic motions 
 of the same period. For the complex number 
 
 where r\, TZ, r^; ai, 02, a* and n are constants may be 
 written 
 
 2 = Zl e nti + zze nti + 
 or
 
 32 GRAPHICAL METHODS. 
 
 z = 2<>e n "', 
 where 
 
 ZO = Zl + 32 + ' ' ' + 2 A . 
 
 The movement of Z therefore, excepting the case z = 0, consists 
 in a uniform revolution of OZ round 0, OZ always keeping the 
 same length equal to the modulus of z . The position of OZ at 
 the moment t = is OZ . 
 The motion of a point P whose abscissa is 
 
 x = ae~ kt cos (id + a) 
 
 where a, k, n, a are constants (a and k positive) is called a damped 
 harmonic motion. It may be looked upon as a harmonic motion, 
 whose amplitude is decreasing. To study this motion let us 
 again substitute a complex number 
 
 z = ae~ kt cos (rd + a) + ae~ kt sin (nt + a)i, 
 or 
 
 z = ae- kt 'eW, 
 or 
 
 z = z <r*<.e" ri , 
 
 where ZQ is written for the complex constant ae**. 
 The product 
 
 is a complex number corresponding to a point Z\ on the same 
 radius as Z , coincident with Z at the moment t = but ap- 
 proaching in a geometrical ratio after t = 0. In unit of time 
 the distance of Zi from decreases in the constant ratio e~ k : 1. 
 The multiplication with e nti turns OZi round through an angle 
 nt. We may therefore describe the motion of Z as a uniform 
 revolution of OZ round 0, Z at the same time approaching 
 at a rate uniform in this sense that in equal times the distance 
 is reduced in equal proportions (Fig. 26). At the moment t = 
 the position coincides with ZQ. We speak of a period of this 
 motion meaning the time T = 2ir/n in which OZ performs an 
 entire revolution round 0, although it does not come back to its
 
 GRAPHICAL CALCULATION. 
 
 original position. Any part of the spiral curve described by Z 
 in a given time is geometrically similar to any other part of the 
 curve described in an interval of equal duration. For suppose 
 the second interval of time hap- 
 pens T units of time later, we 
 shall have for the first interval 
 
 z = z e- kt -e nti , 
 and for the second interval 
 
 Now if zi and Zz are the values 
 of z at two moments ti and k of 
 the first interval and Zi and %' 
 the corresponding values of z' 
 at the moments ti + T and ^ -f T of the second interval, we have 
 
 FIG. 26. 
 
 22 
 
 Therefore the triangle ZiOZ% is geometrically similar to the 
 triangle Zi'OZz'. As Z\ and Zi may coincide with any points 
 of the first part of the curve, the two parts are evidently geo- 
 metrically similar. 
 
 The projection of Z on the axis of x performs oscillations 
 decreasing in amplitude. The turning-points correspond to those 
 points of the spiral curve described by Z, where its tangent is 
 parallel to the axis of y, that is to say, where the abscissa of dz/dt 
 vanishes. 
 
 Now 
 
 dz 
 
 or 
 
 dx
 
 34 GRAPHICAL METHODS. 
 
 where p and X are the modulus and angle of the complex number 
 k + ni. 
 
 Consequently, if we represent dz/dt by a point Z', the triangle 
 Z'OZ will remain geometrically similar to itself. The turning 
 points of the damped oscillations correspond to the moments 
 when OZ r is directed vertically upward or downward or when the 
 angle of dz/dt is equal to r/2 or 37T/2. The angle of z will then 
 be 7T/2 X or 3-7T/2 X plus or minus any multiple of 2w. As 
 the angle of z, on the other hand, is changing in time according 
 to the formula 
 
 rt+a, 
 
 we find the moments where the movement turns by the equation 
 
 nt + a = Tr/2 - X + 2Nir, 
 or 
 
 ni + a = 37T/2 - X + 2Nv, 
 
 N denoting any positive or negative integral number. The time 
 between two consecutive turnings is therefore equal to ir/n, that 
 is, equal to half a period. All the points Z corresponding to 
 turning points lie on the same straight line through the origin 
 forming an angle 3^/2 X with the direction of the positive axis 
 of x. The amplitudes of the consecutive oscillations therefore 
 decrease in the same proportion as the modulus of z, that is 
 to say, in half a period in the ratio e~~Z' 
 
 Let us consider the vibrations of a system possessing one 
 degree of freedom when the system is subjected to a force varying 
 as a harmonic function of the time and let us limit our considera- 
 tions to positions in the immediate neighborhood of a position 
 of stable equilibrium. If the quantity x determines the position 
 of the system the oscillations satisfy a differential equation of 
 the form 
 
 where m, k, n, p, F are positive constants. 
 
 1 See for instance Rayleigh, Theory of Sound, Vol. I, chap. Ill, 46.
 
 GRAPHICAL CALCULATION. 35 
 
 This is another case where the introduction of a complex 
 variable 
 
 z = x + yi 
 
 and the geometrical representation of complex numbers helps to 
 form the solution and to survey the variety of phenomena that 
 may be produced. 
 
 In order to introduce z let us simultaneously consider the 
 differential equation 
 
 and let us multiply the second equation by i and add it to the 
 first. We then have 
 
 The movement of the point Z representing the complex number 
 z then serves as well to show the movement corresponding to x. 
 We need only consider the projection of Z on the axis of x. 
 
 A solution of the differential equation may be obtained by 
 writing 
 
 z = z e pti . 
 
 Introducing this expression for z and cancelling the factor e pti 
 we have 
 
 zo(- mp 2 + kpi + n 2 ) = F, 
 or 
 
 F 
 
 z is a complex constant, that may be represented geometrically 
 as we shall see later on. 
 This solution 
 
 is not general. If z' denotes any other solution so that
 
 36 GRAPHICAL METHODS. 
 
 we find by subtracting the two equations 
 
 or writing 
 
 z' z = u, 
 
 The general solution of this equation is 
 u = u^ + u^t 
 
 where u\ and ^ are arbitrary constants and \\ and X 2 are the 
 roots of the equation for X 
 
 mX 2 + k\ + n 2 = 0, 
 
 If P/4m 2 is greater than n 2 , so that the square root has a real 
 value, V / ^ 2 /4m 2 n z will certainly be smaller than k/2m. There- 
 fore Xi and X 2 will both be negative and the moduli of the complex 
 numbers Ui^ lt and u^*' will in time become insignificant. If, 
 on the other hand, ]t?/4.m 2 is smaller than n 2 , both complex 
 numbers u\f^ 1 * and y^e^ correspond to points describing spirals that 
 approach the origin, as we have seen above, in a constant ratio 
 for equal intervals of time. Therefore they will also in time 
 become insignificant. 
 
 After a certain lapse of time the expression 
 
 z = Zae pti 
 
 will therefore suffice to represent the solution. 
 
 The point Z moves uniformly in a circle round of a radius 
 equal to the modulus of z , completing one revolution in the 
 period 27r/p, the period of the force acting on the system. The
 
 GRAPHICAL CALCULATION. 37 
 
 movement of the projection of Z on the axis of x is given by 
 
 x = r cos (pt + a), 
 
 where TQ is the modulus and a the angle of ZQ. It is a harmonic 
 movement with the same period as that of the force* Fcospt, 
 but with a certain difference of phase and a certain amplitude 
 depending on the values of F, m, k, n, p. 
 
 It is important to study this relation in order to survey the 
 phenomena that may be produced. For this purpose the geo- 
 metrical representation of complex numbers readily lends itself. 
 
 In the expression for ZQ 
 
 F 
 
 let us consider the denominator 
 
 - mp 2 + kpi + 2 , 
 
 and let us suppose the period of the force acting on the system 
 not determined, while the constants of the system m, k, n and 
 the amplitude of the force F have given values. The quantity p 
 is the number of oscillations of the force during an interval of 
 2ir units of time. This quantity p we suppose to be indeter- 
 minate and we intend to show, how the amplitude and phase 
 of the forced vibrations compare with the amplitude and phase 
 of the force for different values of p. 
 
 Let us plot the curve of the points corresponding to the complex 
 number 
 
 n 2 - mp 2 + kpi, 
 
 where p assumes the values p = to +00. 
 
 This curve is a parabola whose axis coincides with the axis of 
 x and whose vertex is in the point x = n 2 , y = 0. We find its 
 equation by eliminating p from the equations 
 
 x = n 2 - mp z , y = kp, 
 viz.,
 
 38 
 
 GRAPHICAL METHODS. 
 
 p-3 
 
 But it is better not to eliminate p and to plot the different points 
 
 for different values of p. In Fig. 27 the curve is drawn for p = 
 
 to 3 and the points for 
 p=Q, 1, 2, 3 are marked. 
 The ordinates increase in 
 proportion to p', they are 
 equal to 0, k, 2k, 3k for 
 p = 0, 1, 2, 3. The dis- 
 tance between the projec- 
 tion of any point of the 
 curve on the axis of x and 
 
 the vertex is proportional to p 2 . It is equal to 0, m, 4m, 9m for 
 
 p = 0, 1, 2, 3. 
 
 For any point P on the parabola let us denote the distance 
 
 from by r and the angle between OP and the positive axis of 
 
 x by <p so that 
 
 n 2 mp 2 + kpi = re**. 
 
 Then we have 
 
 20 = re*', 
 
 and consequently 
 
 FIG. 27. 
 
 2 = - 
 
 and 
 
 = - cos (pt - <f>). 
 
 The amplitude F/r of the forced vibration is inversely propor- 
 tional to r. Thus our Fig. 27 shows us what the period of the 
 force must be to make the forced vibrations as large as possible. 
 It corresponds to the point on the parabola whose distance from 
 is smallest. It is the point where a circle round touches the 
 parabola. In Fig. 27 this point is marked R. It may be called 
 the point of maximum resonance. When the constants of the 
 system are such that the ordinate of the point, where the parabola 
 intersects the axis of y is small in comparison with the abscissa
 
 GRAPHICAL CALCULATION. 39 
 
 of the vertex, then OR will lie close to the axis of y (Fig. 28). In 
 this case the angle between OR and the positive axis of x will be 
 very nearly equal to 90, that is to say, the forced oscillations will 
 lag behind the force oscil- 
 lations by a little less than a 
 quarter of a period. Keep- 
 ing m and n constant, this 
 will take place for small val- 
 ues of k, i. e., for a small 
 damping influence. A small FlG 2 g 
 
 deviation of p from the fre- 
 quency of maximum resonance will throw the point P awayfrom.R, 
 so that r increases considerably and <p becomes either very small 
 (for values of p smaller than the frequency of maximum resonance) 
 or nearly equal to 180 (for values of p larger than the frequency 
 of maximum resonance). In other words for small values of k the 
 maximum of resonance is very sharp. A deviation of the period 
 of the force from the period of maximum resonance will lessen the 
 amplitude of the forced vibration considerably. The lag of its 
 phase behind that of the force will at the same time nearly vanish, 
 when the frequency of the force is decreased or it will become nearly 
 as large as half a period, when the frequency of the force is in- 
 creased. For larger values of k the parabola opens out and this 
 phenomenon becomes less marked. The minimum of the radius r 
 becomes less pronounced. The angle between OR and the axis of 
 x becomes smaller and smaller and for a certain value of k and all 
 larger values the point R will coincide with the vertex of the para- 
 bola. In this case, there is no resonance. When the period of 
 the force increases indefinitely (p becoming smaller and smaller) 
 the amplitude of the forced vibration will increase and will 
 approach more and more to the limit 
 
 
 
 w 2 ' 
 
 but there will be no definite period for which the forced vibra- 
 tions are stronger than for all others.
 
 CHAPTER II. 
 
 THE GRAPHICAL REPRESENTATION OF FUNCTIONS OF ONE OR 
 MOBE INDEPENDENT VARIABLES. 
 
 6. Functions of One Independent Variable. A function y of 
 one variable x 
 
 y = f(*) 
 
 is usually represented geometrically by a curve, in such a way 
 that the rectangular coordinates of its points measured in certain 
 chosen units of length are equal to x and y. This graphical rep- 
 resentation of a function is exceedingly valuable. But there is 
 another way not less valuable for certain purposes, more used in 
 applied than in theoretical mathematics, which here will occupy 
 our attention. 
 
 Suppose the values of y are calculated for certain equidistant 
 values of x, for instance: 
 
 x = - 6, - 5, - 4, - 3, -2, - 1, 0, 
 
 + 1, + 2, + 3, + 4, + 5, + 6, 
 
 and let us plot these values of y in a uniform scale on a straight 
 
 line. Draw the uniform scale on 
 one, side of the straight line and 
 mark the points that correspond 
 to the calculated values of y on 
 the other side of the straight line. 
 Denote them by the numbers x 
 that belong to them (Fig. 29). 
 The drawing will then allow us to 
 read off the value of y for any of 
 FlG 29 the values of x with a certain ac- 
 
 curacy depending on the size of the 
 
 scale and the number of its partitions and naturally on the fine- 
 
 40
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 41 
 
 ness of the drawing. It will also allow us to read off the value 
 of y for a value of x between those that have been marked, if 
 the intervals between two consecutive values of x are so small 
 that the corresponding intervals of y are nearly equal. We can 
 with a certain accuracy interpolate values of x by sight. On the 
 other hand, we can also read off the values of x for any of the 
 values of y. We shall call this the representation of a function 
 by a scale. 
 
 We can easily pass over to the representation of the same 
 function by a curve. We need only draw lines perpendicular 
 to the line carrying the scales through the points marked with 
 the values of x and make their length measured in any given 
 unit equal to the numbers x that correspond to them (Fig. 29). 
 
 In a similar way we can pass 
 from the representation of the 
 function by a curve to the rep- 
 resentation by a scale. 
 
 The representation by a scale 
 may be imagined to signify the 
 movement of a point on a straight 
 line, the values of x meaning the 
 time and the points marked with ^ 
 these values being the positions 
 of the moving point at the times 
 
 marked. By passing over to the curve the movement in the 
 straight line is drawn out into a curve with the time as abscissa 
 (Fig. 30).. 
 
 The representation by a scale is used in connection with the 
 representation by a curve for the purpose of drawing a function 
 of a function. 
 
 Let y be a function of x and x a function of t. Then we wish 
 to represent y as a function of t. 
 
 Let y = f(x) be given by a curve in the usual way and let 
 x = <p(t) be given by a scale on the axis of x marking the points 
 where t = 0, 1, 2, , 12. We then find the values of y corre-
 
 42 
 
 GRAPHICAL METHODS. 
 
 spending to the values t = 0, 1, 2, -, 12 by drawing the ordi- 
 nates of the curve y = f(x) for the abscissas marked t = 0, 1, 2, 
 
 Fio. 31. 
 
 , 12. These ordinates as a rule will not be equidistant. But 
 
 as soon as we move them so as to make them equidistant, they 
 
 * A -PI form the ordinates of the curve 
 
 \ \ 
 
 y = 
 
 with / as abscissa (Fig. 31). 
 
 The representation of a func- 
 tion by a scale may be general- 
 ized in the respect that neither 
 of the two scales facing one an- 
 other on the straight line need 
 necessarily be uniform. The in- 
 tervals of both scales may vary 
 from one side of the scale to the 
 other. If the variation is suffi- 
 
 ciently slow the interpolation can nevertheless be effected with 
 accuracy. We may look at this case as composed of two cases 
 of the first kind. 
 
 f(x) = y and y = g(t}. 
 
 FIG. 32.
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 43 
 
 These scales are placed together, so that the scale x touches the 
 scale t 
 
 /(*) = 0(0, 
 
 while the scale y is cut out (Fig. 32). 
 
 7. The Principle of the Slide Rule. Let us investigate how 
 the relation between x and t changes by sliding the x- and ^-scales 
 along one another. 
 
 If we slide the z-scale through an amount y = c so that a 
 point of the z-scale that was opposite to a certain point y of the 
 ?/-scale, now is opposite y + c, then the relation between x and t 
 represented by the new position of the scales will be given by 
 the equation 
 
 /(*) = 0(0 + c. 
 
 If x, t and x', t', denote two pairs of values that are placed 
 opposite to one another, we shall have simultaneously 
 
 /(*) = g(t) + c, 
 
 /(*') = g(t'} + c, 
 or by eliminating c 
 
 /(*)- 0(0 =/(*') -<7(0- 
 
 The ordinary slide rule carries two identical scales y = log x and 
 y = log t that are able to slide along one another, x and t running 
 through the values 1 to 100. We therefore have 
 
 log x log t = log x' log t', 
 or 
 
 x__ ^ 
 
 t ~~ t' ' . 
 
 FIG. 33. 
 
 That is to say, in any position of the x- and <-scale any two values 
 x and t opposite each other have the same ratio (Fig. 33). This
 
 44 GRAPHICAL METHODS. 
 
 is the principle on which the use of the slide rule is founded. 
 It enables us to calculate any of the four quantities x, t, x', t' 
 if the other three are given. Suppose, for example, x, t, x' 
 known. We set the scales so that x appears opposite to t, 
 
 , r 
 
 
 
 rt i i 
 
 fflSSPi 
 
 20 30 10 ' 60 80 100 
 
 T > } if 
 
 ? f 1 
 
 5 6 j 8 flp 
 
 Y 7 "*" 1 
 
 
 JL5 2 3 
 
 i 5 6 f iUl 
 
 ^ 
 
 
 Tr 
 
 
 
 FIG. 34. 
 
 
 then i' is read off opposite to x'. On the other edges the slide 
 rule carries two similar scales one double the size of the other 
 (Fig. 34). We may write 
 
 y = 2 log X and y = 2 log T. 
 
 By means of a little frame carrying a crossline and sliding over 
 the instrument, we can bring the scales x and T or t and X op- 
 posite each other. If, for example, for any position of the 
 instrument x, T and x', T' are two pairs of values opposite each 
 other, then 
 
 log x - 2 log T = log x' - 2 log 7", 
 or 
 
 x x 
 
 If any three of the four quantities x, T, x', T' are known the 
 fourth may be read off. Thus we find the value 
 
 xT' 2 
 
 by setting T opposite to x and reading off the value opposite to 
 T'. Or we can find the value of 
 
 f-r 
 
 by setting a; opposite to T and reading off the value opposite x'.
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 45 
 
 Let us reverse the part that carries the scales t, T so that x 
 slides along T and X along t, but in the opposite order (Fig. 35). 
 
 FIG. 35. 
 The scales t, T may then be expressed by 
 
 y = I log t and y = I 2 log T, 
 
 I being the entire length of the scales. 
 
 By setting the instrument to any position and considering the 
 scales x and t or X and T by means of the cross line we have 
 log x + log t = log x' + log t' and log X + log T = log X' + log T 
 or 
 
 xt = x't f and XT = XT, 
 
 so that any two values opposite to one another have the same 
 product. 
 
 For x and T we have 
 
 xT 2 = x'T' 2 . 
 Let us apply this to find the root of an equation of the form 
 
 w 3 + au = 6. 
 Divide by u so that 
 
 and set T = 1 opposite to X = 6. Then taking T = u we find 
 on the same cross line t = v? and X = b/u, so that we read the 
 two values w 2 and bfu directly opposite to each other on the 
 scales t and X. If b/u is positive, it decreases while u 2 increases.
 
 46 GRAPHICAL METHODS. 
 
 Running our eye along we have to find the place where the differ- 
 ence b/u u 2 is equal to a. Having found it the T-scale gives 
 us the root of the equation. For example take 
 
 u s - 5u = 3, 
 or 
 
 We set T = 1 opposite X = 3 and run our eye along the scales 
 X and t (Fig. 36), to find the place where i - 5 = X. We find 
 
 
 
 2 TTT 
 
 
 
 
 / ct or oc 
 
 8 
 OT6: 
 
 9 e s ? 
 
 ' ' T 
 
 
 p^- 
 
 
 FIG. 36. 
 
 it approximately at t = 6.2, and on the T-scale we read off 
 T = 2.50 as the approximate value of the root. This is the 
 only positive root. But for a negative root 3/w is negative, 
 and therefore the positive value of 3/w plus w 2 would have to be 
 equal to 5. We run our eye along and find t = 3.37 opposite to 
 X = 1.63, approximately corresponding to T = 1.84. There- 
 fore 1.84 is another root. As the coefficient of u 2 in the first 
 form of the equations vanishes it follows that the sum of the 
 three roots must be equal to zero. This demands a second 
 negative root approximately equal to 0.66. To make sure 
 that it is so, we set the instrument back and take the other end 
 of the T-scale as representing the value T = 1 and give it the 
 position this end had before. Running our eye along the 
 scales X and t, we find t = 0.43 opposite to X = 4.57, giving 
 X + t = 5.00. On the f-scale we find 0.655, so that the third 
 root is found equal to 0.655. 
 
 When b is negative there is always one and only one negative 
 root. For u running through the values u = to oo, u 2 b/u 
 will run from oo to + oo without turning. When b is positive 
 there is always one and only one positive root; for then u? b/u
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 47 
 
 runs from oo to + oo for u = to + oo. In the first case 
 there may be two positive roots or none; in the second case there 
 may be two negative roots or none. For positive values of a 
 one root only exists in either case. This is easily seen in the first 
 form of the equation 
 
 u z + au = b, 
 
 because from a positive value of a it follows that u 3 + au will 
 for M = oo to + oo, run from oo to + without turning 
 and will therefore pass any given value once only. 
 
 In order to decide whether in the case of a negative value of a 
 there are three roots or only one let us write 
 
 U? - = a. 
 u 
 
 For negative values of b we have to investigate whether there 
 are positive roots. For positive values of u the function u?bfu 
 has a minimum, when the differential coefficient vanishes, i. e., for 
 
 Having set our slide rule so that t gives us w 2 and X gives us 
 b/u, we find the value u where the minimum takes place by 
 running our eye along and looking for the values X, t opposite 
 each other for which X is twice the value of t 
 
 2t = X. 
 
 Then t + X is the minimum of u z b/u, so that there will be 
 two or no positive roots according to t + X being smaller or 
 larger than a. For positive values of b, we have to find out 
 whether there are negative roots. The criterion is the same. 
 After having set T = 1 opposite to 6 and having found the
 
 48 GRAPHICAL METHODS. 
 
 positive root, we find the place where 
 2t= X. 
 
 Then t + X is the minimum of all values that u z bfu assumes 
 for negative values of u. If the minimum is smaller than a 
 there are two negative roots; if it is larger there are none. If it 
 is equal to a the two negative roots coincide. 
 For the equation 
 
 for instance, we find t = 1.31 opposite to X = 2.62 (Fig. 36), 
 so that 2t = 2.62 = X. Now t + X = 3.93 is smaller than 5, 
 therefore u 2 3/w will assume the value 5 for two negative 
 values of u on either side of the value u = T = 1.143 
 for which the minimum of u 2 3/w takes place. 
 
 On the same principle as the slide rule many other instruments 
 may be constructed for various calculations. In all these cases 
 we have for any position of the instrument 
 
 /(*)- 0(0 =/(*') -0(O, 
 
 where x, t are any readings of the two scales opposite each other 
 and x't' the readings at any other place. f(x) and g(f) may be 
 any functions of x and t. It will only be desirable that they 
 be limited to intervals of x and t, which contain no turning 
 points. Else the same point of the scale corresponds to more 
 than one value of x or t and that will prevent a rapid reading 
 of the instrument. 
 
 Let us design an instrument for the calculation of the increase 
 of capital at compound interest at a percentage from 2 per cent. 
 upward. If x is the number of per cent, and t the number of 
 years, the increase of capital at compound interest is in the pro- 
 portion
 
 EEPRESENTATION OF FUNCTIONS. 49 
 
 We can evidently build an instrument for which 
 
 For taking first the logarithm and then the logarithm of the 
 logarithm, we obtain 
 
 We have only to make the ar-scale 
 
 y = + loglog(l + j^)- log log (l + 
 
 and the 2-scale 
 
 y = log n log t. 
 
 For x = 2 we have y = and therefore in the normal position 
 of the instrument t n. On the other end we have t = 1 and 
 therefore y = log n. Now let us take n = 100, so that y = 2 
 for t = 1. Say the length of the instrument is to be about 24 
 cm., then the unit of length for the jr-scale would have to be 12 
 cm. In the normal position of the instrument the readings x, t 
 opposite to each other satisfy the equation 
 
 Opposite t = 1, we read the value x\ = 624 and this gives us 
 
 A capital will increase in 100 years at two per cent, compound 
 interest in the proportion 7.24 : 1. Or we may also say the 
 number x\ = 624 read off opposite t = 1 is the amount which is 
 added to a capital equal to 100 by double interest of 2 per cent. 
 in 100 years. The same position of the instrument gives us the 
 number of years that are wanted for the same increase of capital 
 5
 
 50 GRAPHICAL METHODS. 
 
 at a higher percentage. For all the values x, t opposite to each 
 other satisfy the equation 
 
 For any other given percentage x and any other given number 
 of years t the increase of capital is found by setting x opposite 
 to t and reading the z-scale opposite to t = 1. The only restric- 
 tion is that the ratio is not greater than 7.24, else t = 1 will 
 lie beyond the end of the or-scale. 
 
 For a given increase of capital the instrument will enable us 
 either to find the number of years if the percentage is given, or 
 the percentage if the number of years is given, subject only to 
 the restriction mentioned. 
 
 We can build our instrument so as to include greater increases 
 of capital by choosing a larger value of n. n = 1000, for in- 
 stance, will make y = 3 for t = 1. If the instrument is not to 
 be increased in size the scales would have to be reduced in the 
 proportion 2 : 3. 
 
 Let us consider another instance 
 
 1 1 1 
 
 In the normal position of the instrument the scale division 
 marked x = D corresponds to y = and is opposite to t = n. 
 If we have t = oo on the other end, the length of the instrument 
 will correspond to y = 1/n. Let us choose n = 0.1, so that the 
 length of the instrument is y = 10. That is to say, the unit of 
 length of the y-scale is one tenth of the length of the instrument. 
 For any position of the instrument we have 
 
 -+---+- 
 
 x + t~ x'^f 
 
 If the scale division marked x = oo is opposite to t = c we can 
 write x' = oo,t' = c and have
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 51 
 
 1 +- = -. 
 
 X^ t C 
 
 The instrument will therefore enable us to read off any one of 
 the three quantities x, t, c, if the other two are given, the only 
 restriction being that all three lie within the limits 0.1 to oo. 
 The instrument may be used to determine the combined resistance 
 of two parallel electrical re- 
 sistances, for the resistances 
 satisfy the equation 
 
 - 
 R,' 
 
 FIG. 37. 
 
 Similarly it may be used 
 to calculate the distances of an object and its image from the 
 principal planes of any given system of lenses. For if / is the 
 focal length and x and t the distances of the object and its im- 
 age from the corresponding principal planes (Fig. 37), the equa- 
 tion is 
 
 -+-- 1 - 
 
 x + t f 
 
 On the back side of the movable part of an ordinary slide rule 
 there generally is a scale 
 
 y = 2 + log sin t. 
 
 When this part is turned round and the scale is brought into 
 contact with the scale 
 
 y = log x, 
 
 we obtain for any position of the instrument 
 
 log x log sin t = log x' log sin t', 
 
 or 
 
 x x' 
 
 sin t sin t' ' 
 
 for any two pairs of values x, t that are opposite each other.
 
 52 GRAPHICAL METHODS. 
 
 Given two sides of a triangle and the angle opposite the larger 
 of the two the instrument gives at once the angle opposite the 
 other side. Similarly when two angles and one side are given, 
 it gives the length of the other side. 
 
 If x' = a is the value opposite to t' = 90, we have 
 
 x = a sin t. 
 
 Thus we can read the position of any harmonic motion for any 
 value of the phase. 
 
 An instrument carrying the scales 
 
 y log sin x and y = log sin t 
 enables us to find any one of four angles x, t, x', t' for which 
 
 sin x _ sin x' 
 sin t sin t' 
 
 if the other three are given. Thus, knowing the declination, 
 hour angle and height of a celestial body, we can read the azimuth 
 on the instrument. We have only to take x = 90 height, 
 t = hour angle, x' = 90 declination, then t' = azimuth or 
 180 azimuth. 
 
 It is not necessary to carry out the subtraction 90 height and 
 90 declination. The difference may be counted on the scale 
 by imagining written in the place of 90, 10 in the place of 
 80 and so on and counting the partitions of the scale backwards 
 instead of forward. 
 
 8. Rectangular Coordinates with Intervals of Varying Size. 
 The two methods of representing the relation between two 
 variables either by a curve connecting the coordinates or by 
 scales facing each other lead to a combination of both. 
 
 Suppose the rectangular coordinates x and y are functions of 
 u and v, 
 
 x = <f>(u) and y = t(x). 
 
 The function x = <p(u) is represented by a uniform scale for x 
 on the axis of abscissae facing a non-uniform scale for u. The
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 53 
 
 function y = ^(c) is represented by a uniform scale for y on 
 the axis of ordinates facing a non-uniform scale for v. Through 
 the scale-divisions u let us draw vertical lines, and through the 
 scale-divisions v let us draw horizontal lines. These two systems 
 of parallel lines form a network of rectangular meshes of various 
 sizes (Fig. 38), and any equation between u and v may be repre- 
 sented by a curve in this plane. 
 
 The usefulness of this method will be seen by some examples. 
 It enables us by a clever choice of the functions <p(u) and $(v) 
 
 > 
 
 FIG. 38. 
 
 123 i 5 
 FIG. 39. 
 
 to simplify the form of the curve. It is easily seen, for instance, 
 that a curve representing an equation /(w, v) = may always be 
 replaced by a straight line, if we choose the w-scale properly. 
 For when the points u = 1, 2, 3, 4, of the curve are not on 
 a straight line, let them be moved to a straight line without 
 altering their ordinates (Fig. 39). This will change the w-scale 
 but it will not alter the equation f(u, ) = now represented by 
 the straight line. 
 
 Suppose we want to represent the relation 
 
 where a and 6 are given numbers. If u and v were ordinary 
 rectangular coordinates the curve would be an ellipse. But if we 
 make 
 
 x = u 2 and v = v* 

 
 54 GRAPHICAL METHODS. 
 
 the equation of the line in rectangular coordinates becomes 
 
 and the curve will therefore be a straight line running from a 
 point on the positive axis of x to a point on the positive axis of 
 y. The point on the axis of x corresponds to the value u = =*= a 
 
 on the w-scale, and the 
 point on the axis of y cor- 
 responds to the value v = 
 =t b on the r-scale (Fig. 40). 
 Any point on the straight 
 line corresponds to four 
 combinations -j-w, +0; u, 
 + v, u, v; u, v, be- 
 cause x has the same values 
 for opposite values of u 
 We can read v as a function of 
 
 2.5 
 2 
 
 1.5 
 1 
 
 -J.-.L.O _* irf.a io rj 
 
 FIG. 40. 
 
 and y for opposite values of 
 u or u as a function of v. 
 If a second equation 
 
 is given, we find the common solutions of the two equations by 
 the intersection of the corresponding straight lines. Fig. 40 
 shows the solutions of the two equations 
 
 1 
 
 and 
 
 * 2V 
 
 42 -I- 52 ~ 
 
 approximately equal to u = =*= 1.2 and = 2.4. 
 Another function much used in mathematical physics 
 
 _ 
 
 v = ae m *
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 55 
 
 may also be represented by a straight line by means of the same 
 device. 
 By making 
 
 y log v, x = u 2 , 
 we obtain 
 
 where log v and log a are the natural logarithms of v and a. 
 The i/-scale is laid off on the axis of x and the u-scale on the axis 
 of y and we have to join the points u = 0, v = a and u = m, 
 v = a/e. The point v = a/e is found by laying off the distance 
 v = 1 to v = e from v = a downward (Fig. 41). We are not 
 obliged to take the same units of length for x and y. 
 
 FIG. 41. 
 
 Suppose we had to find the constants a and m from two equa- 
 tions 
 
 and 
 
 _o 
 
 = ae ^' 
 
 Our diagram would furnish two points corresponding to u\, vi 
 and u%, TZ. The straight line joining these two points intersects 
 the axis of ordinates at v = a and intersects the parallel through 
 x = a/e to the axis of abscissae at u = m.
 
 56 
 
 GRAPHICAL METHODS. 
 
 In applied mathematics the problem would as a rule present 
 itself in such a form that more than two pairs of values u, v 
 would be given but all of them affected with errors of observation. 
 The way to proceed would then be to plot the corresponding 
 points and to draw a straight line through the points as best we 
 can. A black thread stretched over the drawing may be used to 
 advantage to find a straight line passing as close to the points 
 as possible (Fig. 42). 
 
 In several other cases the variables u and v are connected with 
 the rectangular coordinates x and y by the functions 
 
 x = log u and y = log v. 
 
 4 ' 
 
 Fio. 43. 
 
 "Logarithmic paper" prepared with parallel lines for equidistant 
 values of u and lines perpendicular to these for equidistant values 
 of v is manufactured commercially (Fig. 43). 
 
 By this device diagrams representing the relation 
 
 u r v* = c, 
 
 where r, s, c are constants are given by straight lines. For by 
 taking the logarithm we obtain 
 
 rx + sy = log c. 
 
 The straight line connects the point u = e 1/r on the w-scale with 
 the point v = c l/t on the fl-scale. 
 
 Logarithmic paper is further used to advantage in all those
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 57 
 
 cases where a variety of relations between the variables u and v 
 are considered that differ only in u and v being changed in some 
 constant proportion. If u and v were plotted as rectangular 
 coordinates the curves representing the different relations be- 
 tween u and v might all be generated from one of them by altering 
 the scale of the abscissae and independently the scale of the ordi- 
 nates, so that the appearance of all these curves would be very 
 different. Let us write 
 
 M ) = o, 
 
 as the equation of one of the curves. The equations of all the 
 rest may then be written 
 
 where a, b are any positive constants. The points u, v of the 
 first curve lead to the points on one of the other curves by taking 
 u a times as great and v b times as great. For if we write u' = au 
 and v' = bu the equation f(u, v) = leads to the equation 
 between u' and v': 
 
 Using logarithmic paper the diagram of all these curves be- 
 comes very much simpler. The equation /(w, v) = is equivalent 
 to a certain equation <p(x, y) = 0, where x = log u, y = log i>. 
 Now let x', y' be the rectangular coordinates corresponding to 
 u', v' so that 
 
 x' log u' = log u + log a = x + log a, 
 y' = log v' = log v + log 6 = y + log b. 
 
 The point x', y' is reached from the point x, y by advancing 
 through a fixed distance log a in the direction of the axis of x 
 and a fixed distance log b in the direction of the axis of y. The 
 whole curve 
 
 f(u, v) =
 
 58 GKAPHICAL METHODS. 
 
 drawn on logarithmic paper is therefore identical with all the 
 
 It can be made to coincide with any one of the curves by 
 moving it along the directions of x and y. 
 
 9. Functions of Two Independent Variables. When a func- 
 tion of one variable y = f(x) is represented by a curve, the values 
 of x are laid off on the axis of x and the values of y are represented 
 by lines perpendicular to the axis of x. In a similar way a 
 function of two independent variables 
 
 * = /(*, y} 
 
 may be represented by plotting x and y as rectangular coordinates 
 and erecting lines perpendicular to the xy plane, in all the 
 points x, y, where f(x, y) is defined and making the lengths of 
 the perpendiculars proportional to 2. In this way the function 
 corresponds to a surface in space* Now there are practical 
 difficulties in working with surfaces in space and therefore it 
 appears desirable to use other methods, that enable us to represent 
 functions of two independent variables on a plane. This may 
 be done in the following way. 
 
 Taking x, y as rectangular coordinates all the points for which 
 f(x, y) has the same value form a curve in the xy plane. Let 
 us suppose a number of these curves drawn and marked with the 
 value of f(x, y}. If the different values of f(x, y) are chosen 
 sufficiently close, so that the curves lie sufficiently close in the 
 part of the xy plane that our drawing comprises, we are not only 
 able to state the value of f(x, y} at any point on one of the drawn 
 curves, but we are also able to interpolate with a certain degree 
 of accuracy the value of f(x, y} at a point between two of the 
 curves. As a rule it will be convenient to choose equidistant 
 values of f(x, y) to facilitate the interpolation of the values 
 between. The curves may be regarded as the perpendicular 
 projection of certain curves on the surface in space, the inter-
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 59 
 
 sections of the surface by equidistant planes parallel to the 
 xy plane. 
 
 The method is the generalization of the scale-representation 
 of a function of one variable. For a relation between t and x 
 represented by a curve with t as ordinate and x as abscissa, is 
 transformed into a scale representation by perpendicularly 
 projecting certain points of the curve onto the axis of x, the 
 intersections of the curve by equidistant lines parallel to the axis 
 of x and marking them with the value of t. A scale division in 
 the case of a function of one variable corresponds to a curve in 
 the case of a function of two independent variables. 
 
 This method of representing a function of two independent 
 variables by a plane drawing or we might also say of representing 
 a surface in space by a plane drawing, is used by naval architects 
 to render the form of a ship and by surveyors to render the form 
 of the earth's surface and by engineers generally. Let us apply 
 the method to a problem of pure mathematics. 
 
 The equation q- -p?_ -i. 3 
 
 + pz + q = 
 
 defines z as a function of p and q. Let us represent this function 
 by taking p and q as rectangular coordinates and drawing the 
 lines for equidistant values of z. 
 
 For any constant value of z we have a linear equation between 
 the variables p and q, and therefore it is represented by a straight 
 line. This line intersects the parallels p = 1 and p = 1 at 
 the points q = z 3 z and q = z 3 + z. Let us calculate 
 these values for z = 0; 0.1; 0.2 1.3 and in this way 
 draw the lines corresponding to these values of z as far as they 
 lie in a square comprising the values p = 1 to + 1 and 
 q = 1 to + 1. Fig. 44 shows the result. On this diagram 
 we can at once rsad the roots of any equation of the third degree 
 of the form 
 
 z 3 + pz + q = 0, 
 
 where p and q lie within the limits 1 to + 1. For p = 0.4 and
 
 60 
 
 GRAPHICAL METHODS. 
 
 q 0.2, for instance, we read z = 0.37, interpolating the value 
 of z according to the position of the point between the lines 
 z = 0.3 and z = 0.4. We also see that there is only one real 
 root, for there is only one straight line passing through the point. 
 
 -1.3 
 
 0.3 
 
 FIG. 44. 
 
 On the left side of the square there is a triangular-shaped region 
 where the straight lines cross each other. To each point within 
 this region corresponds an equation with three real roots. For 
 example, at the point p= 0.8 and q = + 0.2 we read z = 
 1.00; + 0.28; + 0.72. On the border of this region two roots 
 coincide. 
 
 For values of p and q beyond the limits 1 to + 1 the diagram 
 may also be used. We only have to introduce z' = z/ra instead 
 of z and to choose m sufficiently large. 
 
 Instead of 
 
 z 3 + pz + q =
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 61 
 
 we obtain 
 
 mV 3 + pmz' + q = 0, 
 or dividing by m 3 , 
 
 where 
 
 *'-& '- 
 
 By choosing a sufficiently large value of m, p' and g' can be 
 made to lie within the limits 1 to -+- 1 so that the roots z' 
 may be read on the diagram. Multiplying them by m we 
 obtain the roots z of the given equation. 
 
 A function of two independent variables need not be expressed 
 in an explicit form, but may be given in the form of an equa- 
 tion between three variables 
 
 g(u, 0, w} = 0, 
 
 and we may consider any two of them as independent and the 
 third as a function of the two. The graphical representation 
 may sometimes be greatly facilitated by modifying the method 
 described before. The curves for constant values of one of the 
 three variables, say w, are not plotted by taking u and v as 
 rectangular coordinates, but they are plotted after introducing 
 new variables x and y, x a function of u and y a function of v and 
 making x and y the rectangular coordinates. 
 
 In some cases, for instance, we can succeed by a right choice 
 of the functions x = <p(u) and y = $(v) in getting straight lines 
 for the curves w = const. This will evidently be the case, 
 when the equation g(u, v, w) = can be brought into the form 
 
 a(w~)<p(u) + b(w}\l/( / e) + c(u') = 0, 
 
 a, b, c being any functions of w, <f> any function of u and \f/ any 
 function of v. 
 For introducing
 
 62 GRAPHICAL METHODS. 
 
 * = <t>(u), y = t(v) 
 the equation will become 
 
 ax + by + c = 0, 
 
 where a, b, c are constants for any constant value of w. 
 
 As an example let us consider the relation between the true 
 solar time, the height of the sun over the horizon, and the declina- 
 tion of the sun for a place of given latitude. Instead of the 
 declination of the sun we might also substitute the time of the 
 year, as the time of the year is determined by the declination of 
 the sun. Our object then is to make a diagram for a place of 
 given latitude from which for any time of the year and any 
 height of the sun the true solar time may be read. 
 
 In the spherical triangle formed by 
 the zenith Z, the north pole P (if we sup- 
 pose the place to be on the northern 
 hemisphere) and the sun S (Fig. 45), the 
 sides are the complements of the decli- 
 nation 6, the height h, and the latitude 
 <p. The angle t at the pole is the hour 
 angle of the sun, which expressed in 
 time gives true solar time. 
 
 The equation between these four quantities may be written in 
 the form 
 
 sin h = sin <p sin 5 + cos <f> cos 5 cos t. 
 
 The latitude <p is to be kept constant, so that t, h, 5 are the only 
 variables. 
 Now let us write 
 
 x = cos t, y = sin h, 
 so that the equation takes the form 
 
 y = sin <p sin 8 + % cos <p cos 5. 
 When x and y are plotted as rectangular coordinates, we obtain
 
 GKAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 63 
 
 a straight line for any value of 5. Let us draw horizontal lines 
 for equidistant values of h = to 90 and vertical lines for equi- 
 distant values of t = 180 to + 180 or expressed in time 
 from midnight to midnight (Fig. 46). In order to draw the 
 
 latitude = O. 
 
 FIG. 46. 
 
 straight lines 8 = const., let us calculate where they intersect 
 the vertical lines corresponding to x = 1 and x = + 1 or 
 expressed in time corresponding to midnight and to noon. For 
 x = 1 we have y = cos (<p + 6), and for x = + 1 we have 
 y = cos (<p 5). Let us draw a scale on the vertical x = 1 
 showing the points y = cos (p + 5) for equidistant values of 
 (<p + 6) and a scale on the vertical x = + 1, showing the points 
 y cos (<f> 5) for equidistant values of <p 8. The scale is 
 the same as the scale for h, with the sole difference that the values 
 of <p 8 are the complements of h and the values of <p + 8 the 
 complements of h. For a latitude of 41, for instance, we 
 
 have 
 
 For S <p + s <? - d 
 
 June 21 23.5 64.5 17.5 
 
 September 23 and March 21 41 41 
 
 December 21 -23.5 
 
 17.5 
 
 64.5 e
 
 64 
 
 GRAPHICAL METHODS. 
 
 The values of <p + 5 and <p 6 furnish the intersections with 
 the verticals x = 1 and x = + 1, so that the straight lines 
 can be drawn corresponding to these days of the year. The two 
 outward lines are parallel but the middle line is steeper. Their 
 intersections with the horizontal line h show the time of 
 sunrise and sunset. 1 Strictly speaking the straight lines do 
 not correspond to certain days. The straight line determined 
 by any value of 8 changes its position continually as 5 changes 
 continually. But the changes of 5 during one day are scarcely 
 appreciable unless the drawing is on a larger scale. 
 If in the equation 
 
 ax + by -f c = 
 
 a and b are independent of w, only c being a function of w, all 
 the straight lines w = const, are parallel. In this case we are 
 
 not obliged to draw the 
 straight lines w = const. 
 It will suffice to draw a 
 line perpendicular to the 
 lines w = const, and a 
 scale on it that marks the 
 points corresponding to 
 equidistant values of w. 
 On the drawing we place a 
 sheet of transparent paper 
 or celluloid,on which three 
 straight lines are drawn is- 
 suing from one point in the direction perpendicular to the w-scale, 
 7>-scale and w-scale (Fig. 47). If we move the transparent material 
 without turning it and make the first two lines intersect the u-and-v 
 scale at given points, the w-scale will be intersected at the point 
 corresponding to the value of w. This method has the advantage 
 
 1 That is to say, the moment when the center of the sun would be seen on 
 the horizon, if there were no atmospherical refraction. To take account of 
 the refraction, the line h = 0.6 would have to be considered instead of 
 h = 0. 
 
 FIG. 47.
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 65 
 
 that we can use the same paper for a great many relations of 
 three variables, as we can place a great many scales side by side. 
 Or, in the case of one relation only, we may divide the region of 
 the values u, v, w into a number of smaller regions and draw three 
 scales for each of them, placing all the w-scales or ^-scales or 
 w/'-scales side by side. The drawing will then have the same 
 accuracy as a drawing of very much larger size in which there 
 is only one scale for each of the three variables. 
 
 10. Depiction of One Plane on Another Plane. Let us now 
 consider two quantities x and y each as a function of two other 
 quantities u and v 
 
 x = <p(u, r), 
 
 y = \l/(u, T). 
 
 In order to give a geometrical meaning to this relation between 
 two pairs of quantities let us consider x and y as rectangular 
 coordinates of a point in a plane and u, v as rectangular coordi- 
 nates of a point in another plane. We then have a corre- 
 spondence between the two points. When the functions (p(u, rs) 
 and \lt(u, v) are defined for the values u, v of a certain region, 
 they will furnish for every point u, v of this region a point in 
 the xy plane. Let us call this a depiction of the uv plane on 
 the xy plane. Similarly a function of one variable x = <p(u) 
 might be said to depict the u line on the x line. We may there- 
 fore say that the depiction of one plane on another plane is, in 
 a certain way, the generalization of the idea of a function of one 
 variable. Let us suppose <p(u, v) and \f/(u, v) both to have only 
 one value for given values of u and v for which they are defined. 
 Then there will be only one point in the xy plane corresponding 
 to a given point in the uv plane. But to a given point in the 
 xy plane there may very well correspond several points in the 
 uv plane. 
 
 Let us try to explain this by a graphical representation of the 
 depiction of planes on each other. For this purpose we draw 
 the curves x = const, and y = const, in the uv plane for equi-
 
 66 
 
 GRAPHICAL METHODS. 
 
 distant values of x and y. In the xy plane they correspond to 
 equidistant lines parallel to the axis of x and to the axis of y. 
 The point of intersection of two lines x = a and y = b corre- 
 sponds to the points of intersection of the curves 
 
 <f>(u, v) = a and \l/(u, v) = b, 
 
 in the uv plane. If in a certain region of the uv plane, that 
 we consider, they intersect only once there is only one point in 
 the region of the uv plane considered and one point in the xy 
 plane corresponding to each other. Fig. 48 shows the depiction 
 of part of the uv plane on part of the xy plane. We have a net 
 of square-shaped meshes in the xy plane and corresponding is a 
 net of curvilinear meshes in the uv plane. 
 
 Let us consider the curves x = const, in the uv plane as the 
 perpendicular projections of curves of equal height on a surface 
 extended over that part of the uv plane. From any point P 
 of the surface corresponding to the values u, v we proceed an 
 
 iro.7 
 
 0.2 0.3 0.4 0.5 0.6 
 
 FIG. 48. 
 
 infinitely small distance, u changing to u -f- du, v to v + dv and 
 x to x + <&c, where 
 
 7 dv> , , d<p , 
 
 Let us write 
 
 du = cos ads, dv = sin ads, 
 
 where ds signifies the length of the infinitely small line from 
 u, v to u + du, v + dv in the uv plane and a the angle its direc-
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 67 
 
 tion forms with the positive axis of x. Let PN be a straight line 
 whose projections on the u and v axis are equal to d<p/du and 
 d(p/dv and let us write 
 
 -=rcosX, ^-= 
 
 r being the positive length of PN and X the angle between its 
 direction and the positive axis of x. Then we have 
 
 dx = du + dv = rds cos (a X), 
 
 (jit uV 
 
 or 
 
 dx 
 
 dx/ds measures the steepness of the ascent. It is positive when 
 the direction leads upward and negative when it leads downward 
 and its value is equal to the tangent of the angle of the ascent. 
 From the equation 
 
 ~r = r cos (a X) 
 
 we see that the ascent is steepest for a = X, where dx/ds = r. 
 The line PN in the u, 0-plane shows the perpendicular projection 
 of the line of steepest ascent on the surface x = <p(u, v) and the 
 length of PN measured in the same unit of length in which u and 
 v are measured is equal to the tangent of the angle of the ascent. 
 Let us call the line PN the gradient of the function <p(u, v) at the 
 point u, v. The direction of the gradient is perpendicular to the 
 curve <p(u, v) = const, that passes through the point u, v; for in 
 the direction of the curve we have 
 
 = 
 
 and therefore 
 
 a - X = 90. 
 
 If PN' is the gradient of the function $(u, v) at the point u, v, the 
 angle between PN and PN' must be equal to the angle formed
 
 68 GRAPHICAL METHODS. 
 
 by the curves x = const, and y = const, that intersect at the 
 point u, v, or equal to its supplement according to the angle of 
 intersection that we consider. 
 
 Suppose the gradients PN and PN' do not vanish in any of 
 the points in the region of the uv plane that we consider and 
 that their length and direction vary as continuous functions of 
 u and v. Let us further suppose that the gradient PN' (com- 
 ponents: d\f//du, d^/dfl) is for the whole region on the left side 
 of the gradient PN (components: d<p/du, d<p/dv), or else for the 
 whole region on the right side of the gradient PN, then it fol- 
 lows that any one of the curves x = const, and any one of the 
 curves y = const, can only intersect once in the region considered. 
 
 This may be shown by considering the directions of the curves 
 x == const, and y = const, in the uv plane. Let us consider 
 that direction on the curve y = const, in which x increases. If 
 this direction deviates from PN the deviation must be less than 
 90, because dx/ds and therefore cos (a X) is positive. Let us 
 further consider that direction on the curve x const, in which 
 y increases. If it deviates from the direction of PN' the devia- 
 tion must be less than 90. Let us call these directions the 
 direction of x (on the curve y = const.) and the direction of y 
 (on the curve x = const.). Now if the gradient PN' is on the 
 left of the gradient PN the y direction must also be on the left 
 of PN (for if it were on the right of PN being perpendicular to 
 PN it would form an obtuse angle with PN') and therefore it 
 must be on the left of the x direction (for if it were on the right, 
 PN' being perpendicular to the x direction would form an obtuse 
 angle with the y direction, which we have seen to be impossible). 
 Similarly it may be seen, that if PN' is on the right of PN, the 
 direction of y will also be on the right of the direction of x. If 
 therefore PN' is on the same side of PN in the whole region 
 considered, the direction of y will also be on the same side of the 
 direction of x for the whole region considered. This excludes 
 the intersection of two curves x = const, and y const, in more 
 than one point. For, suppose there are two points of inter-
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 69 
 
 section and we pass along the curve y = const, in the direction of 
 x. At the first point of intersection we pass over the curve 
 x = const, from the side of smaller values of x to the side of 
 larger values of x. Now if the values of x go on increasing 
 as we go along the curve y = const, we evidently cannot get 
 back to a curve x = const, corresponding to a smaller value of x. 
 The only possibility of a second point of intersection would be 
 that the direction in which the value of x increases on the curve 
 y = const, becomes the opposite, so that in advancing in the 
 same direction in which we came x would decrease again. 
 
 The same holds for the curve 
 x= const. If we pass from one 
 point of intersection with a 
 curve y = const, along a curve 
 x = const, to a second point 
 of intersection with the same 
 curve the only possibility is 
 that the direction of y also be- 
 comes opposite. This is ex- 
 cluded as in contradiction with FlQ> 49> 
 the direction of y being on the 
 same side of the direction of x throughout the whole region (Fig.49) 
 
 It will be useful to look at it from another point of view. Let 
 us consider a point A in the uv plane corresponding to the 
 values u, v and let us increase u and v by infinitely small positive 
 amounts du and dv, so that we get four points ABCD, forming a 
 rectangle corresponding to the coordinates. 
 
 A : u, v, B : u + du, v; C : u, v -f- dv; D : u -f- du, v + dv. 
 
 In the xy plane these points are depicted in the points A, 
 B, C, D, the intersections of two curves u and u + du with two 
 curves v and v + dv (Fig. 50). 
 
 The projections of the line AB in the xy plane on the axes of 
 coordinates are obtained by calculating the changes of x and y 
 for a constant value of v and a change du in the value of u
 
 70 
 
 GRAPHICAL METHODS. 
 
 Similarly the projections of AC are obtained by calculating the 
 changes of x and y for a constant value of u and a change dv in the 
 value of V 
 
 , d<p. . d#, 
 <fe= -^dv, dy 2 =~dv. 
 
 Denoting the lengths of AB and AC by d$i and ds z and the angles 
 that the directions of AB and AC form with the direction of the 
 
 dv 
 
 FIG. 50. 
 
 positive axis of x (the angles counted in the usual way) by 71 
 and 72 we have: 
 
 dxi = dsi cos 71, dyi = dsi sin 71 
 
 and 
 or 
 
 and 
 
 We may call 
 
 = ds 2 cos 72, dyz = ds z sin 72, 
 
 d<p 
 
 the scale of depiction at A in the direction AB and
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 71 
 
 dv 
 
 the scale of depiction at A in the direction AC. It is here under- 
 stood that the uv plane is the original, which is depicted on the 
 xy plane. If we take it the other way the scales of depiction 
 in the directions AB and AC are the reciprocal values dulds\ 
 and dvfds-i. 
 
 The area of the parallelogram ABCD in the xy plane is 
 
 / \ 
 
 an fe - 70 - - ^ 
 
 According to the way in which the angles 72 and 71 are defined 
 sin (72 ~~ 7i) is positive, when the direction AC points to the left 
 of the direction AB (assuming the positive axis of y to the left 
 of the positive axis of x), and sin (72 71) is negative, when AC 
 points to the right. Now dudv is equal to the area of the rectangle 
 ABCD in the uv plane. Therefore the value of 
 
 du dv dv du 
 
 is the ratio of the areas ABCD in the two planes and its positive 
 or negative sign denotes the relative position of the directions 
 AB and AC in the xy plane. We may call this ratio the scale 
 of depiction of areas at the point A. 
 
 du dv dv du 
 
 is called the functional determinant of the functions p(u, v) and 
 t(u, ). 
 
 We have found the scale of depiction of lengths in the direc- 
 tions AB and AC. Let us now try to find it in any direction 
 whatever. From any point A in the uv plane, whose coordinates 
 are u and x, we pass to a point D close by whose coordinates are 
 u + Aw, x + At>. In the xy plane we find the corresponding 
 points A and D with coordinates (Fig. 51).
 
 72 
 
 GRAPHICAL METHODS. 
 
 ^ . x = <p(u, v) p. x + Ax = ??(M -f- AM, u + Au) 
 y = &(u. c) ' y ~r~ Aw = ^(M ~j~ AM, ?? -|- A??) 
 
 We expand according to Taylor's theorem, and writing for 
 shortness 
 
 d & 3 <p dw $\1/ 
 
 we find 
 
 Ax = ^AM + <f>v&v + terms of higher order, 
 
 Ay = faAu + faAv + terms of higher order. 
 
 Au 
 
 Fio. 61. 
 
 The length of AD and the angle of its direction we denote by 
 Ar and a in the uv plane and by A* and X in the xy plane. 
 The limit of the ratio A*/Ar, to which it tends, when D approaches 
 A without changing the direction AD is the scale of depiction 
 at the point A in the direction AD. 
 Writing 
 
 AM = Ar cos a, 
 Az> = Ar sin a, 
 we obtain 
 
 Aa: = (<f> u cos a + <p v sin a)Ar + terms of higher order, 
 Ay = (^u cos a + fa sin a)Ar + terms of higher order. 
 
 Dividing by Ar and letting Ar decrease indefinitely, we have in 
 the limit 
 
 dx 
 
 --= (p u cos a + V* sin a,
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 73 
 
 dy 
 j- = \f/ u cos a + ^ v sin a. 
 
 For dx/dr and <ft//dr we ma y also write dsfdr cos X, ds/dr sin X. 
 
 & 
 
 T" cos X = > M cos a + < sin a, 
 
 ds . 
 
 T" sin X = ^ u cos a -{- \f/ v sin a. 
 
 These equations show the scale of depiction ds/dr corresponding 
 to the different directions X in the x, y-plane and a in the u, v- 
 plane. 
 
 By introducing complex numbers we can show the connection 
 still better. 
 Let us denote 
 
 dx dy . ds x< 
 2 -j- + -r 1 = T e *, 
 dr r dr dr 
 
 Zl = Vu + tui, 
 22 = * + &0. 
 
 Multiplying the second of the two equations by i and adding 
 both they may be written as one equation in the complex form: 
 
 2 = Zi cos a -f- Zz sin a. 
 
 The modulus of z is the scale of depiction of the wo plane at the 
 point A in the direction a. The angle of z gives the direction in 
 the xy plane corresponding to the direction a. For a = we 
 have z = Zi and for a = 90, z = 22. 
 Let us substitute 
 
 COS a 
 
 2 2t 
 
 and write 
 
 2 ' 
 so that the expression for z becomes
 
 74 GRAPHICAL METHODS. 
 
 z = ae ai + be~ ai . 
 
 This suggests a simple geometrical construction of the complex 
 numbers z for different values of a. The term ae ai is represented 
 by the points of a circle described by turning the line that 
 represents the complex number a round 
 the origin through the angles a=Q- -2ir. 
 The term be~ ai is represented by the 
 points of a circle described by turning 
 the line that represents b round the ori- 
 gin in the opposite direction through the 
 angles a = - 2ir (Fig. 52). The 
 addition of the two complex numbers 
 jijg 62 ae ^ i an d be ai for any value of a is easily 
 
 performed. The points corresponding 
 
 to the complex numbers z describe an ellipse, whose two princi- 
 pal axes bisect the angles between a and b. This is easily seen 
 by writing 
 
 a = ne (a<r - ai)i , b = r 2 e (ao+ai)i . 
 
 a Q corresponds to the direction bisecting the angle between a 
 and b and i denotes half the angle between a and 6 (positive or 
 negative according to the position of a and 6). 
 
 = fa + r 2 ) cos (a ai) + fa r 2 ) sin (a aji. 
 
 Denoting the coordinates of the complex number ze~ aoi by and 77 
 we have 
 
 = sin (a ai), 
 and consequently the equation of an ellipse 
 
 fa + r 2 ) 2 ~*~ fa - r 2 ) 2 = 1 '
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 75 
 
 This ellipse turned round the origin through an angle equal to 
 o gives us the points corresponding to z. The principal axes 
 are 2(n + r 2 ) and 2(n - r 2 ) (Fig. 53). The construction of 
 
 FIG. 53. 
 
 Fig. 53 is obvious. After plotting z\ and Zz we find Zz/i and 
 Zz/i by turning AZ 2 through a right angle to the right and to 
 the left. From these points lines are drawn to Z\. The bisection 
 of these lines give a and b. 
 
 The figure shows that in case a and 6 have the same modulus, 
 the triangle Zz/i, Z\, Zz/i becomes equilateral and AZ\ is per- 
 pendicular to the line joining Zz/i and Zj/t. In this case AZ\ 
 and AZz would have the same or the opposite direction. But as 
 zi = <f>u + tui, Zz = <f>v + iM, this would mean that (f> u \f/ v <p v ^ u 
 = 0. 
 
 The radii of the ellipse (Fig. 53) measured in the unit used 
 give the different scales of depiction corresponding to the dif- 
 ferent directions in the xy plane. We might also say the ellipse 
 is the image in the xy plane of an infinitely small circle in the 
 uv plane, magnified in the proportion of the infinitely small radius 
 to 1, with its center in A. 
 
 Zi corresponds to a = and Z 2 to a = 90 and for a = to 90
 
 76 
 
 GRAPHICAL METHODS. 
 
 Z moves on the ellipse from Z\ to Zz through the shorter way. 
 Zi corresponds to a = 180 and Zz to a = 270. Now we 
 have shown above that a positive value of the functional deter- 
 minant put* Vvtu means that Zz is on the positive side of Z\, 
 so that in this case Z moves in the positive sense (that is, in the 
 direction from the positive axis of x to the positive axis of y) with 
 increasing values of a. With a negative value Z moves in the 
 opposite direction. 
 
 Let us now suppose that the curves x = const, and y = const, in 
 the uv plane intersect except on a certain curve where their direc- 
 
 -y, 
 
 -1/4 
 
 u 
 FIG. 54. 
 
 tions coincide in the way shown in Fig. 54. On this curve the 
 functional determinant D = ip u ^ v <p v ^ u must vanish because 
 the directions of the gradients coincide. Let us see what the 
 depiction on the xy plane is like. 
 
 Running along one of the curves y = const., say y = y\, 
 toward the curve D = we intersect the curves x = x, x$, Xz 
 until at the point A on the curve x = x t we reach the curve D = 0. 
 In the xy plane the corresponding path is a parallel to the axis 
 of x at a distance y\ passing through #4, z 3 , Xz and reaching a 
 point A at XL If we now proceed on the curve y = yi in the 
 uv plane beyond the curve D = 0, we again intersect the curves 
 Xz, x 3 , etc., but in the inverse order. Thus the corresponding 
 path in the xy plane does not pass beyond A, but turns back
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 77 
 
 through the same points z 2 , y\\ x 3 , y 1} etc. The same holds for 
 any of the other lines y = const. If we trace the line in the 
 xy plane that corresponds to the points in the uv plane, where 
 the curves x = const, and y = const, touch, we find the depiction 
 of the uv plane only on one side of the curve in the xy plane. 
 The other side has no corresponding points u, v. However to 
 every point C on this side of the curve, there are two correspond- 
 ing points C in the uv plane, one on either side of the curve 
 D = 0. Imagine two sheets of paper laid on the xy plane; let 
 them both be cut along the curve AB. Retain only the two 
 pieces on this side of the curve and paste them together along 
 the curve. The uv plane is in this way depicted on the paper 
 in such a way that there is one point and one only on the paper 
 corresponding to each point in 
 the region of the uv plane con- 
 sidered. The curve D = in 
 the uv plane corresponds to the 
 rim where the two pieces of pa- 
 per are pasted together. Any 
 line straight or curved passing FIG. 55. 
 
 over the curve D = in the uv 
 
 plane,corresponds to a line running from one of the sheets onto the 
 other. It need not change its direction abruptly when it reaches 
 the rim and passes onto the other sheet. For it may touch the 
 rim in the direction of its tangent. This is actually the rule 
 and the abrupt change of direction is the exception. Any line 
 LAL (Fig. 55) in the uv plane, whose tangent as it crosses the 
 curve D = at A does not coincide with the common tangent 
 of the curves x = const, and y = const, will correspond to a line 
 in the xy plane, that does not change its direction abruptly 
 when it touches the rim. 
 
 This is best understood analytically. Let us consider corre- 
 sponding directions at the points A in the uv plane and in the 
 xy plane. We have seen above that corresponding directions 
 (Fig. 56) are connected by the equations
 
 78 
 
 GRAPHICAL METHODS. 
 
 dy/dr- 
 
 dx/dr 
 
 FIG. 56. 
 
 <& dx 
 
 cos X y~ = -j" = <f> u cos a + <f> v sin a, 
 
 ds dy 
 - = -7- = ^u cos a + ^ c sm a. 
 
 At the point A we have 
 
 Assuming that the gradients at A do not vanish, so that we 
 can write 
 
 <p u = r cos 7 , <p v = r sin 7, 
 
 \f/ u = r' cos 7', \l/ v = r' sin 7', 
 
 where r and r' are positive quantities, the equation (p u ^ v v^u=0 
 reduces to sin (7 7') = 0, that is, 7 = 7' or 7 = 7'+ 180. 
 It follows therefore that: 
 
 ds 
 cosX^ = r cos (a 7), 
 
 sin XT" = r' cos (a 7') = =*= r' cos (a 7). 
 
 Consequently for all directions a. in the uv plane for which 
 cos (a 7) is not zero, we have 
 
 tgX
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 79 
 
 That is to say, we have in the xy plane only one fixed direction 
 X and the opposite corresponding to all the different directions 
 a except only a direction for which cos (a 7) = 0. In the 
 latter case, that is, when the direction a is perpendicular to the 
 direction 7 of the gradient, i. e., in the direction of the curves 
 x = const, and y = const., we have 
 
 ds 
 
 ds 
 
 sin X -j- = 0. 
 dr 
 
 Therefore ds/dr = and X remains indeterminate. Any direction 
 X for which tg X differs from + r'/r corresponds to a fixed direction 
 a = y + 90 or a = y - 90, while ds/dr = 0. 
 
 As the curve D = is depicted on the rim of the two sheets 
 of paper, all those lines that intersect the curve D = in a 
 direction different from the direction of the curves x = const. 
 and y = const, are depicted in the xy plane as curves having 
 their tangent at A in common with the rim. All lines in one of 
 the sheets of paper that touch the rim at A in a direction differ- 
 ent from that of the rim must be the depiction of lines in the wo 
 plane that reach A in the direction of the lines x = const, and 
 y = const. The scale of depiction is zero in the direction of the 
 curves x = const, and y = const. In any other direction a 
 we find it different from zero for: 
 
 = V(? + r ' 2 ) cos 2 (a - T ). 
 
 It is a maximum in the direction a = y or y + 180 perpendicular 
 to the curves x = const, and y = const. 
 
 It may help to understand all these details if we discuss an 
 example where the depiction of the uv plane on the xy plane 
 has a simple geometrical meaning, the planes being ground plan 
 and elevation of a curved surface in space. The rim in the 
 xy plane is the outline of the surface, the projection of those
 
 80 
 
 GRAPHICAL METHODS. 
 
 points where the tangential plane is perpendicular to the plane 
 
 of elevation. 
 
 Suppose a cylinder of circular section cut in two half cylinders 
 
 by a plane through its axis. Suppose one of the half cylinders 
 
 in such a position that its axis 
 forms an angle 8 with the 
 ground plan, the plan of ele- 
 vation being parallel to its 
 axis, Fig. 57. Let us intro- 
 duce rectangular coordinates 
 u, v in the ground plan and 
 rectangular coordinates x, y 
 in the plan of elevation. A 
 point P on the cylinder is de- 
 fined by certain values u, v 
 which define its ground plan 
 and certain values x, y which 
 define its elevation. It is 
 easily seen from Fig. 57 that 
 we have 
 
 AB 
 
 V 
 
 A 
 
 R 
 
 X 
 
 D 
 
 .1 
 
 P 
 
 4 l 
 
 u 
 
 ( 
 
 
 \l \ 
 
 
 and 
 
 where a is the radius of the section. Now let us consider the 
 elevation of the points P as a depiction of their ground plan. 
 The functions <p(u, v) and \f/(u, ) in this case are 
 
 <p(u, v) = u, 
 \fs(u, v) = u tg d + 
 
 and 
 
 
 
 1 
 cos 6 
 
 = tgS, 
 
 cos 5 I/a 2 
 
 cos 5 I/a 2
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 81 
 
 CD 
 
 FIG. 58. 
 
 The functional determinant vanishes for v = on the line EF. 
 The lines y = const, are the intersections of the cylinder with 
 horizontal planes. In the plan of 
 elevation they are straight hori- 
 zontal lines; in the ground plan 
 they are ellipses (Fig. 58). As we 
 pass along one of these curves we 
 cross the line EF in the ground 
 plan but we only touch it in the 
 plan of elevation, retracing the hori- 
 zontal line back again. The lines 
 x = const, are straight lines in both 
 planes, but in space they corre- 
 spond to ellipses. Again as we 
 cross EF in the ground plan we 
 only touch it in the plan of eleva- 
 tion and retrace the vertical line down again. Any curve on 
 the cylinder that crosses EF in a direction not perpendicular to 
 the plan of elevation is projected in the plan of elevation with 
 EF as its tangent. For the real tangent in space lying in the 
 tangential plane of the cylinder can have no other projection, if 
 not perpendicular to the plan of elevation. In this latter case 
 the projection of the tangent is a point 
 and the tangent of the elevation is deter- 
 mined by the inclination of the osculatory 
 plane. 
 
 There is a particular case to be consid- 
 ered, when the curve D = in the uv plane 
 coincides with one of the curves x = const, 
 or y = const. (Fig. 59), assuming the gra- 
 dients of the functions <p(u, v) and ^(u, v) 
 not to vanish at the points of this curve. We have seen that at 
 a point where D = the scale of depiction must vanish in the 
 directions of the curve x = const, or y = const. Let the curve 
 D = coincide with a line x = const., then it follows that the 
 
 7 
 
 D=o 
 
 FIG. 59.
 
 82 
 
 GRAPHICAL METHODS. 
 
 length of the depiction of this curve is zero and the depiction 
 must be contracted in a point. For the length of the depiction 
 of a curve x = const, is given by an integral 
 
 where dr denotes an element of the curve and ds/dr the scale 
 of depiction in the direction of the curve. As ds/dr is zero all 
 along the curve the integral must necessarily vanish. 
 
 As an example let us con- 
 sider 
 
 X = UV, 
 
 y = v. 
 
 The lines x = const, in the uv 
 * plane are equilateral hyper- 
 bolas, the lines y = const, are 
 parallels to the axis of u (Fig. 
 60). Along the axis of u we 
 have at the same time y 0, 
 x= and D= v = 0. The 
 whole axis of u is depicted in 
 the point x = 0, y of the xy plane. 
 
 Let us finally consider the case where the scale of depiction 
 at any point is the same in all directions, though it need not be 
 the same at different points. 
 Writing as before 
 
 FIG. 
 
 dx , dy . ds .. 
 
 3 = T + :r* = T e 
 dr dr dr 
 
 the connection between the scale of depiction ds/dr and the 
 angles X, a determining corresponding directions in the xy plane 
 and in the uv plane is given by the equation 
 
 Z = z\ cos a + Zz sin a,
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 83 
 
 or 
 where 
 
 In the case where the scale of depiction ds/dr, that is to say, the 
 modulus of z, is independent of a, one of the constants a or b 
 must vanish, as we see at once from the construction of z (Fig. 
 52). Let us consider the case 6 = 0, 
 
 ,---.". 
 
 dr 
 
 The complex number a may be written \ a \ e^, where | a \ 
 denotes the modulus of a and a the angle. Both may vary 
 from point to point, but at every point they have fixed values. 
 Consequently we have 
 
 -j = | a | and X = a + . 
 
 That is to say, from an angle a determining a direction in the 
 uv plane, we find the angle X determining the corresponding 
 direction in the xy plane by the addition of a fixed value a - 
 Any two directions a, a' will therefore form the same angle as 
 the corresponding directions X, X' in the xy plane. The same is 
 true when a = and z = 6e~ a< . The only difference is that in 
 this latter case the direction of z rotates in the opposite sense 
 with increasing values of a. 
 
 Analytically depictions of this kind are represented by func- 
 tions of complex numbers, 
 
 x + yi = f(u + m} or x + yi = f(u m). 
 
 Assuming the function to possess a differential coefficient we have 
 
 dx dy. ^ 
 
 fr
 
 84 
 
 GRAPHICAL METHODS. 
 
 and therefore either 
 
 Hence in the first case 
 a = |(zi + Z 
 and in the second case 
 
 or 
 
 a = 0, b = ZL 
 
 11. Other Methods of Representing Relations between Three 
 Variables. The depiction of one plane on another may be used 
 to generalize the graphical representation of a function of two 
 variables or a relation between three variables, as we prefer 
 to say. 
 
 As we have seen before, an equation 
 
 g(x, y, z) = 
 
 between three variables x, y, z can be represented by taking x 
 and y as rectangular coordinates and plotting the curves z = 
 const. (Fig. 61) for equidistant val- 
 ues of z. Suppose now the xy plane 
 to be depicted on another plane. 
 The lines x = const., y = const, and 
 z = const, will be represented by 
 three sets of curves. The fact that 
 three values x, y, z satisfy the equa- 
 tion g(x, y, z) = is shown geo- 
 metrically by the intersection of 
 the three corresponding curves in 
 one point. 
 
 Another method for representing certain relations between 
 three variables u, v, w consists in drawing three curves, each 
 curve carrying a scale. The values of u, v, w are read each on 
 one of the three scales. The relation between three values u, v, 
 w is represented geometrically by the condition that the corre- 
 sponding points lie on a straight line (Fig. 62). This method is 
 
 FIG. 61.
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 85 
 
 far more convenient than the one using three sets of curves. It is 
 less trouble to place a ruler over two points u, v of two curves 
 and read the value w on the scale of the third than to find the 
 intersection of two curves u = const, and v = const, among sets 
 of others, pick out the curve w = const, that passes through the 
 
 same point and read the value of w corresponding to it. For we 
 must consider that the curves corresponding to certain values 
 of u and v are generally not drawn, but must be interpolated and 
 so must the curve w = const. It is true that interpolations are 
 necessary with both methods, but the interpolation on scales 
 like those in Fig. 62 is easily done. 
 
 It must however be understood that while the three sets of 
 curves form a perfectly general method for representing any rela- 
 tion between three variables, the other method is restricted to cer- 
 tain cases. In order to investigate this subject more fully we 
 shall have to explain the use of line coordinates. 
 
 When we apply rectangular coordinates x, y to define a certain 
 point in a plane, we may say that x determines one of a set of 
 straight lines (parallel to the axis of ordinates) and y determines 
 one of another set of straight lines (parallel to the axis of abscissas) 
 and the point is the intersection of the two (Fig. 63, /). A 
 similar method may be used to determine a certain straight line 
 in a plane. Let x determine a point on a certain straight line, 
 x being its distance from a fixed point A on the line measured 
 in a certain unit and counted positive on one side and negative 
 on the other. Let y define a point on another straight line
 
 86 
 
 GRAPHICAL METHODS. 
 
 parallel to the first, y being its distance from a fixed point B on 
 the line measured in the same way as x. The straight line 
 passing through the two points is thus determined by the values 
 
 <D y 
 
 (XI) .a 
 
 FIG. 63. 
 
 x and y and for all possible values of x and y we obtain all the 
 straight lines of the plane except those parallel to the lines on 
 which x and y are measured. For simplicity we choose AB 
 perpendicular to the two lines (Fig. 63, 77). Let us call x and y 
 the line coordinates of the line connecting the two points x and 
 y in Fig. 63, 77, in the same way as x and y in Fig. 63, 7, are 
 called the point coordinates of the point where the two lines 
 x and y intersect. 
 
 A linear equation between point coordinates 
 
 y = mx + n 
 
 is the equation of a straight line. That is to say, all the points 
 whose coordinates satisfy the equation lie on a certain straight 
 line. If, on the other hand, we regard x and y as line coordinates 
 we find the analogous theorem: all the straight lines whose 
 line coordinates satisfy the equation 
 
 y = mx + ju 
 
 pass through a certain point. The equation is therefore called 
 the equation of the point. 
 
 In order to show this let us first draw the line x = 0, y = n 
 (APO in Fig. 64). If now for any value of x we make AR = x
 
 GRAPHICAL REPRESENTATION" OF FUNCTIONS. 
 
 87 
 
 and PQ = mx, the point of intersection of RQ and AP must be 
 independent of x, for 
 
 PO 
 
 Af\ 
 
 AO 
 
 mx 
 
 - 
 
 x 
 
 The ratio PO/AO determines the position of and as it is 
 independent of x and the positions of A and P are also inde- 
 pendent of x, the same is true for 0. 
 For negative values of m, PO and 
 AO have opposite directions so that 
 lies between A and P. 
 
 For a given point 0, we can find 
 the corresponding values of m and M 
 by joining with the points A and 
 the point corresponding to x = 1. 
 If P and Q are the intersections of 
 these lines with the line on which y 
 is measured, we have BP = M and PQ = m. Any point in the 
 plane thus leads to an equation 
 
 y = mx + n, 
 
 except the points on the line on which x is measured. For 
 m = the equation reduces to 
 
 FIG. 64. 
 
 that is, the equation of a point on the line on which y is measured. 
 
 Instead of y = mx + /z, we might also write x = m'y + /*', 
 and go through similar considerations changing the parts of x 
 and y. This form does not include the points on the line on 
 which y is measured, but it does include the points on the line 
 on which x is measured. For these we have m' = 0. 
 
 The general equation of a point in line coordinates is given in 
 the form 
 
 ax + by + c = 0, 
 
 from which we may derive either of the first-mentioned forms 
 dividing it by a or 6.
 
 GRAPHICAL METHODS. 
 
 Dividing by c another convenient form is obtained, 
 
 c c 
 or writing 
 
 c _ c 
 
 a * 6 
 
 , 
 
 ZO 2/0 
 
 x determining the point of intersection of the line BO (Fig. 64) 
 and the a>line, while 3/0 determines the point of intersection of 
 the line AO with the y-line. 
 
 A curve may be given by an equation 
 
 ai(u}x + bi(u)y + ci(u) = 0, 
 
 in which ai(u), bi(u}, c\(u) are functions of a variable u. Any 
 value of u furnishes the equation of a certain point and as u 
 changes the point describes the curve. Let us suppose the curve 
 drawn and a scale marked on it giving the values of u in certain 
 intervals sufficiently close to interpolate the values of u be- 
 tween them. Two other curves are in the same way given by 
 the equations 
 
 02(20* + bz(v)y + <%(*) = 0, 
 
 (w) = 0, 
 
 and scales on these curves mark the values of v and w. 
 
 Now we are enabled to formulate the condition which must be 
 satisfied by the values u, v, w in order that the three corresponding 
 points lie in one straight line. If x and y are the line coordinates 
 of the line passing through the three points, x and y must satisfy 
 all three equations simultaneously. 
 
 Consequently the determinant of the three equations must 
 vanish 
 
 &iC 3 ) + as(biC 2 6ad) = 0, 
 and, vice versa, if the equation between u, v, w may be brought 
 

 
 GHAPHICAL REPRESENTATION OF FUNCTIONS. 8S 
 
 into this form where a\, bi, c\ are any functions of u, 02, b%, GZ any 
 functions of v and a$, 63, c$ any functions of w, we can form the 
 equations 
 
 a\x+ hy + ci = 0, 
 
 d2X + Ihy + cz = 0, 
 + c 3 = 0, 
 
 and represent them graphically by curves carrying scales for 
 u, v, w. The relation between u, v, w is then equivalent to the 
 condition that the corresponding points on the three curves lie 
 on a straight line. But it must be remembered that only a 
 restricted class of relations can be brought into the required form, 
 so that the method cannot be applied to any given relation. 
 The equation of a point 
 
 ax + by -f- c = 
 
 remains of the same form, when the units of length are changed 
 for x and y. If x' denotes the number measuring the same length 
 as the number x but in another unit, the two numbers must have a 
 constant ratio equal to the inverse ratio of the two units. There- 
 fore, by changing -the units independently, we have 
 
 x = \x', y = py', 
 and the equation of the point may be written 
 
 d\x f + buy' + c = 0, 
 or 
 
 a'x' + b'y' + c = 0, 
 
 where a' = Xa and b' nb. 
 
 It is sometimes convenient to define the line coordinates ir 
 another way. Let and 77 denote rectangular coordinates 
 measured in the same unit, then the equation of a straight line 
 can be written 
 
 77 = tg <p + 770, 
 
 where <p is the angle between the line and the axis of and 770
 
 90 GRAPHICAL METHODS. 
 
 the ordinate of the point of intersection with the axis of ?;. 
 Now let us call tg <p and 770 the line coordinates of the straight 
 line represented by the equation and let us denote them by x 
 and y. Thus the values of x and y define a certain straight line 
 and any straight line not parallel to the axis of ordinates may 
 be defined in this manner. The condition that a straight line 
 x, y passes through a point , 77 is expressed by the equation 
 
 i] = a* + y, 
 or 
 
 y = & + 17- 
 
 If we fix the values of x and y, all the values , rj that satisfy this 
 equation represent the points of the straight line x, y and we 
 therefore call it the equation of the straight line. If, on the 
 other hand, we fix the values of and 77, all the values x, y that 
 satisfy the equation represent the straight lines that pass through 
 the given point |, TI, and therefore we call it the equation of the 
 point. 
 
 The more general form 
 
 ax + by + c = 
 can be reduced to 
 
 fe- 
 lt therefore represents the equation of the point, whose rec- 
 tangular coordinates are = a/6 and rj = c/b. The case 
 where 6 = or 
 
 ax + c = 
 
 represents the equation of a point infinitely far away in the 
 direction <f> or the opposite direction <p -\- 180, <p being defined by 
 
 tg*< 
 
 All the straight lines, whose coordinates x, y satisfy the equation 
 ax + c =
 
 GEAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 91 
 
 correspond to the same value of x but to any value of y. That 
 is to say, they are all parallel and all the straight lines of this 
 direction belong to them. 
 
 Let us now discuss some of the applications of line coordinates 
 to the graphical representation of relations between three 
 variables. 
 
 The relation 
 
 uv = 
 
 may be written in the form 
 
 log u + log v = log w, 
 
 or 
 
 when 
 
 x = log u 
 
 x + y = log w, 
 
 and y = 
 
 FIG. 65. 
 
 Let us plot x and y as line co- 
 ordinates on two parallel lines (Fig. 
 65), with scales for the values of u 
 and v. The equations x = log u 
 and y = log v may be regarded as the equations of the points of 
 these two scales. The equation 
 
 x + y = log w 
 
 for any value of w is the equation of a point. It can easily be 
 constructed as the intersection of any lines x, y satisfying its 
 equation. For instance, the line x = log w, y = and the line 
 = 0, y = log w. The first line is found by connecting the 
 scale division u = w of the w-scale with the point B, the second 
 by connecting the scale division v = w of the 0-scale with the 
 point A. If the units of x and y are taken of the same length, the 
 point of intersection will lie in the middle between the two lines 
 carrying the u and v scales on a line parallel to the two other lines 
 and the w-scale will be half the size of the other two (Fig. 65).
 
 92 
 
 GRAPHICAL METHODS. 
 
 The relation 
 
 log u + log v = log w 
 expresses the condition that the three equations 
 
 x = log u, y = log t>, x + y = log w 
 
 are satisfied simultaneously by the same values of x and y, that 
 is to say, that the three points on the u, v, w scales corresponding 
 to the values pf u, v, w lie on the same straight line x, y. 
 The more general relation 
 
 where a. and /3 are any given values, can be treated in the same 
 manner. Thus the pressure and volume of a gas undergoing 
 adiabatic changes may be represented. In this case we have 
 
 pv k = w, 
 
 where p denotes the pressure, v the volume and k and w con- 
 stants. 
 
 For a given gas k has a given value, but w depends on the 
 quantity of the gas considered. 
 
 We write 
 
 x = log p, y = log v. 
 The relation then takes the form 
 
 x + ky = log w, 
 
 and represents a point which may be con- 
 structed by the intersection of any two 
 i straight lines x, y, whose coordinates sat- 
 isfy the equation, for instance 
 
 and 
 
 x = log w, y = 
 
 1 
 
 =, y =
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 93 
 
 +0.5 
 
 +1.0 
 
 +0.5 
 
 The first line connects the point B (Fig. 66) with the scale 
 division p = w of the p scale and the second line connects the 
 point A with the scale division of the v scale for which y = k log w. 
 A perpendicular from the point of intersection on AB meets it in 
 0' and as the ratio AO'/O'B is 
 equal to the ratio of the seg- 
 ments on the p and v scales 
 log w/k log w = l/k it is inde- 
 pendent of w. All the points 
 corresponding to different val- 
 ues of w lie on the same par- 
 allel to the p and v scales and 
 the w scale may be obtained 
 by a central projection of the 
 p scale on this parallel from 
 the center B (Fig. 66). We 
 might dispense with the con- 
 struction of the w scale as 
 long as the straight line for 
 the w scale is drawn. For in 
 using the diagram we gener- 
 ally start with values po, VQ 
 and want to find other values 
 p, v, for which 
 
 pv k = poi)o k . FlG - 67 - 
 
 The straight line connecting the scale divisions p and v intersects 
 the w scale at the same point as the straight line connecting the 
 scale divisions p and v , so that we need not know the value of 
 poVo k . It suffices to mark the point of intersection in order to 
 find the value of p, when v is given or the value of v when p is 
 given. 
 
 Another example is furnished by the equation 
 
 -1.0 
 
 -1.5 
 
 -0.5 
 
 -LO 
 
 -1.5 
 
 -2.0
 
 94 GRAPHICAL METHODS. 
 
 If we regard x and y as line coordinates any value of w determines 
 the equation of a point. We plot the curve formed by these 
 points with a scale on it indicating the corresponding values of w. 
 Any values of x and y determine a straight line whose inter- 
 sections with the w scale furnish the roots of the equation. Each 
 point of the w scale may be constructed by the intersection of 
 two straight lines, whose coordinates x, y satisfy the equation, 
 for instance 
 
 x = 0, y = it? . and x = w, y = O. 1 
 
 In Fig. 67 the w scale is shown for the positive values w = to 
 w = 2.5. 
 
 In the same manner a diagram for the solution of the cubic 
 equation 
 
 w* + xw + y = 0. 
 
 or of any equation of the form 
 
 w* -f xw* + y = 
 may be constructed. 
 
 12. Relations between Four Variables. The method can be 
 generalized for relations between four variables. 
 
 Suppose four variables u, v, w, t are connected by the equation 
 
 g(u, v, w, t) = 0, 
 
 and let us assume that for any particular value t = t the resulting 
 relation between u, v, w can be given by a diagram of the form 
 considered consisting of three curves carrying scales for u, v and 
 w. Let us further suppose that for other values of t the scales 
 for u and v remain the same, but the scale for w changes. Then 
 we shall have a set of w scales corresponding to different values 
 of t. Connecting the points that correspond to the same value 
 of w we obtain a network of curves t = const, and w = const. 
 (Fig. 68). Any two values u, v furnish a straight line intersecting 
 1 For small values of w, this combination is not good because the angle of 
 intersection is small. One might substitute x = 2, y = V? 2w for the 
 first line.
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 
 
 95 
 
 the network of curves. The points of intersection correspond to 
 values of t and w that satisfy the given relation. 
 Any relation of the form 
 
 v(u)f(t, w) + t(v)g(t, w) + h(t, w) = 
 
 may be represented in this way, <p(u} denoting any function of 
 u, ^(r) any function of v andf(t, 
 w), g(t, w}, h(t, w} any functions 
 of t and w. 
 
 In this case we need only in- 
 troduce the line coordinates x, 
 y, writing 
 
 x = v(u), y = t(v). 
 
 We then obtain a linear equation 
 between x and y, 
 
 f(t, w)x + g(t, w)y + h(t, w) = 0, 
 
 which for any given values of t 
 and w represents the equation of 
 
 a point. For a given value of t and variable values of w we obtain 
 a curve t = const, carrying a scale for w and for a series of values 
 of t we obtain a set of curves t = const. Similarly for a given 
 value of w and variable values of t the equation furnishes a curve 
 w = const., carrying a scale for t and a series of values of w 
 furnishes a set of curves w = const. From any given values 
 of u and v the line coordinates x and y are calculated and the 
 points where this straight line defined by x and y intersects 
 the network of the curves t = const, and w = const, furnish 
 the values t, w that satisfy the relation together with the given 
 values of u and v. The relation between the height, azimuth, 
 declination of a celestial body and the latitude of the point of 
 observation may serve as an example. Let h, a, 8 denote the 
 height, azimuth and declination and v the latitude. The angles 
 7T/2 <p, 7T/2 h, 7T/2 d are the three sides of a spherical
 
 GRAPHICAL METHODS. 
 
 Fio. 69. 
 
 triangle PZS (Fig. 69) formed by the pole P, the zenith Z and the 
 celestial body S. The azimuth is defined as the supplement of 
 the angle PZS. 
 The equation is 
 
 sin 5 = sin <p sin h cos <p cos h cos a. 
 
 We write 
 
 x = cos a, y = sin 5, 
 
 so that the equation becomes 
 
 y = sin #> sin h x cos ^ cos h. 
 
 We shall in this case use the second system of line coordinates 
 
 where x is the slope of the line measured by the tangent of the 
 
 angle formed with the axis of 
 
 abscissas and y is the ordinate 
 
 of the intersection with the axis 
 
 of ordinates. If , ij denote the 
 
 rectangular coordinates of the 
 
 point, the equation of the points 
 
 takes the form 
 
 n = x + y or y = 17 x, 
 so that in our case we have 
 = cos <p cos h, i}= sin tp sin h. 
 
 The curves <f> = const, and h = 
 
 const, can be drawn by means 
 
 of these formulas. It is easily 
 
 seen that they are ellipses and 
 
 that the curves <p = const, are 
 
 the same as the curves h = const. 
 
 For a definite value of <p and a j^ 70 
 
 variable value of h we find 
 
 . 
 
 I
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 97 
 
 and for a definite value of h and a variable value of <p 
 
 cos 2 h sin 2 h 
 
 Any of the ellipses intersects all the others and in this way they 
 form a network. A point of intersection of the ellipse <p = c\ 
 and the ellipse h = c 2 also corresponds to the values h = c { and 
 <p = c 2 , as the ellipse <p = Ci is identical with the ellipse h = c\ 
 and <p = GZ identical with h = eg (Fig. 70). The easiest way to 
 find this network consists in drawing the straight lines 
 
 + i] = cos (<p k) } 
 
 and perpendicular to them the straight lines 
 | 77 = cos (<p + h}, 
 
 for equidistant values of <f> + h and <p h. The ellipses run 
 diagonally through the rectangular meshes formed by the two 
 systems of straight lines. The scales for <p and h are written 
 on the axis of coordinates, both scales being available for both 
 variables. The scale for 5 is written on the axis of ordinates 
 and is identical with the scale for t and h on this axis. For the 
 ordinate corresponding to a given value 5 = c is sin c, and this is 
 also the ordinate of the point where the ellipse <p = c or h = c 
 intersects the axis of ordinates. The scale for the azimuth cannot 
 be laid down in exactly the same way as that for <?, h and 5 
 because cos a determines the slope of the straight line x, y. 
 Let us draw a parallel to the axis of ordinates through the point 
 = 1, ?/ = and mark a scale for the azimuth on it, making 
 77 = cos a (Fig. 70). A line connecting the origin with any scale 
 division of this scale has the slope of the line x = cos a, y = sin 5. 
 To bring it into the position of the line x, y it must be moved 
 parallel to itself, until its point of intersection with the axis of 
 ordinates coincides with the scale division 5. This suggests 
 another way of using the diagram. Let a pencil of rays be 
 drawn from the origin to the scale divisions of the azimuth scale 
 (Fig. 70), and let it be drawn on a sheet of transparent paper 
 
 8
 
 98 GRAPHICAL METHODS. 
 
 placed over the drawing of the ellipses. For any given value 
 of 8 it is moved up or down as the case may be so that the center 
 of the pencil coincides with the scale division 5. As long as the 
 celestial body does not materially alter its declination the dia- 
 gram in this position will enable us to find any of the three 
 values <p, h, a from the other two. 
 
 As a second example let us consider the relation between the 
 declination 8, the azimuth a, the hour angle t of a celestial body 
 and the latitude p of the point of observation. 
 
 The relation is found by eliminating the height h from the 
 equation 
 
 sin 8 = sin <p sin h cos <p cos h cos a. 
 
 For this purpose we express sin h and cos h by the other angles 
 and substitute these expressions for sin h and cos h. 
 We have 
 
 cos h = cos 8 sin t/sm a, 
 
 sin h = sin <f> sin 8 + cos <f> cos 8 cos t. 
 
 Substituting these values we find 
 
 sin 8 = sin 2 <p sin 8-\- sin tp cos <p cos 5 cos t cos <p cos 8 sin t ctg a, or 
 
 cos 2 <p sin 8 = sin <p cos <p cos 5 cos t cos <p cos 8 sin t ctg a. 
 Dividing by cos 2 <f> cos 8 we finally obtain 
 
 tg 6 = tg <f> cos t ctg a. 
 
 COS <p 
 
 In order to represent this relation graphically we introduce line 
 coordinates 
 
 x = ctg a and y = tg 8 
 and find 
 
 sin t 
 
 y=tg*cos*- x. 
 
 Let us use the second system of line coordinates. The rec- 
 tangular coordinates , tj of the point represented by the equation 
 are found from it equal to :
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS. 99 
 
 sin t 
 
 The curves <p = const, are ellipses, 
 
 The curves t = const, are hyperbolas, 
 
 sin 2 1 cos 2 1 
 
 1. 
 
 The ellipses and hyperbolas are confocal, the foci coinciding 
 with the points = 1, t\ = 0, so that the curves intersect at 
 right angles. 
 
 The scale for <p may be written on the axis of ordinates at the 
 points where it intersects the ellipses. It is identical with the 
 scale for 6, the ordinate in both cases 
 being the tangent of the angle with the 
 only difference that 5 is negative on 
 the negative part of the axis and <p is 
 not. The scale for t may be written 
 on one of the ellipses corresponding to 
 the largest value of <p that is to be taken 
 account of. This ellipse forms the 
 boundary of the diagram, so that 
 larger values of p are not represented. 
 Corresponding to the azimuth we draw 
 a pencil of rays on a sheet of trans- 
 parent paper, which is laid on the draw- 
 ing of the curves. The center of the 
 pencil is placed on the scale division 5 
 and the azimuth is equal to the angles 
 
 that the rays form with the positive direction of the axis of or- 
 dinates (Fig. 71). It suffices to draw the curves and the rays 
 only on one side of the axis of ordinates. At the apex of the 
 
 FIG. 71.
 
 100 GRAPHICAL METHODS. 
 
 hyperbolas the value of t changes abruptly. The line t = Q h is 
 meant to start from the focus = 1, i) = 0. When the center of 
 the pencil of rays is in the origin the rays form the asymptotic 
 lines of the hyperbolas, a = 15 corresponding to t = l h , a = 30 
 to t = 2 h and so on. 
 
 GIFT OF 
 DR. GEORGE F. McEWEN
 
 CHAPTER III. 
 
 THE GRAPHICAL METHODS OF THE DIFFERENTIAL AND 
 INTEGRAL CALCULUS. 
 
 13. Graphical Integration. We have shown how the ele- 
 mentary mathematical operations of adding, subtracting, multi- 
 plying and dividing and the inverse operation of finding the 
 root of an equation can be carried out by graphical methods and 
 how functions of one or more variables may be represented and 
 handled. But the graphical methods would lack generality and 
 would be of very limited use, if they were not applicable to the 
 infinitesimal operations of differentiation and integration. In- 
 deed it is here that they are found of the greatest value. In 
 many cases, where the calculus is applied to problems of natural 
 science or of engineering, the functions concerned are given in a 
 graphical form. Their true analytical structure is not known 
 and as a rule an approximation by analytical expressions is not 
 easily calculated nor easily handled. In these cases it is of vital 
 importance that the operations of the calculus can be performed, 
 although the functions are only given graphically. 
 
 Let us begin with integration, because it is easier than differ- 
 entiation and of more general application. 
 
 Suppose a function y = f(x) given by a curve whose ordinate is 
 y and whose abscissa is x. The problem is to find a curve, whose 
 ordinate Y is an integral of the function f(x), 
 
 Let us assume the unit of length for the abscissas independent 
 of the unit of length for the ordinates. The value of Y measures 
 the area between the ordinates corresponding to a and x, the 
 curve y = f(x) and the axis of x in units equal to the rectangle 
 formed by the units of x and y. 
 
 101
 
 102 
 
 GRAPHICAL METHODS. 
 
 In the simple case where f(x) is a constant the equation 
 = f(x) = c is represented by a line parallel to the axis of x and 
 
 Y = I cdx = c(x a). 
 
 7 is the ordinate of a straight line intersecting the axis of x at 
 the point x = a. The constant c is the change of Y for an 
 
 increase of x equal to 1. 
 
 v If P is the point on the 
 
 axis of x for x = 1 and 
 Q the point where the line 
 y = c intersects the axis 
 of ordinates (Fig. 72) the 
 desired line is parallel to 
 
 FlQ 72 PQ. It is constructed by 
 
 drawing a parallel to PQ 
 
 through the point x = a on the axis of x (Fig. 72, where a = 0). 
 When a given value c\ is added, so that the equation becomes 
 
 Y = c (x- a) + ^ 
 
 it amounts to the same as when the straight line is moved in the 
 direction of the axis of ordinates through a distance c\. For 
 x = a we then have Y = c\, so that we obtain the line 
 
 Y = c(x - a) + ci, 
 
 by drawing a parallel to PQ through the point x = a, y = ci. 
 
 In the second place let us assume that the line y = f(x) consists 
 of a number of steps, that is to say, that the function has different 
 constant values in a number of intervals x = x\ to Xz, x% to #3, 
 etc., while it changes its value abruptly at 3%, x 3 , etc. The 
 line presenting the integral 
 
 F = 
 
 does not change its ordinate abruptly. It consists of a con- 
 tinuous broken line, whose corners have the abscissas x 2 , x 3 , etc.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 103 
 
 The directions of the different parts are found in the way just 
 described by the pencil of rays from P to the points a, ft, y, etc. 
 (Fig. 73), where the horizontal lines intersect the axis of ordinates. 
 
 FIG. 73. 
 
 To construct the broken line we draw a parallel to Pa through 
 the point x = Xi (in Fig. 73 x\ is equal to 0) as far as the vertical 
 x = x 2 . Through the point of intersection with the vertical 
 x = x z we draw a parallel to P/3 as far as the vertical x = x 3 . 
 Through the point of intersection with the vertical x = x 3 we 
 draw a parallel to Py and so on. 
 
 Finally let us consider the case of an arbitrary function y = f(x) 
 represented by any curve. In order to find the curve 
 
 we substitute for y = f(x) a function consisting of different 
 constant values in different intervals and changing its value 
 abruptly when x passes from one interval to the next, so that 
 the line representing this function consists of a number of steps 
 leading up or down according to the increase or decrease of f(x}. 
 These steps are arranged in the following way. The horizontal
 
 104 GRAPHICAL METHODS. 
 
 part AiA 2 of the first step (Fig. 73) starts from any point AI 
 of the given curve. The vertical part A 2 Bi and the following 
 horizontal part BiB 2 are then drawn in such a manner that BiB 2 
 intersects the curve and that the integral of the given function 
 as far as the point of intersection KI, is equal to the integral of the 
 stepping line as far as the same point. That is to say, the areas 
 between the stepping line and the given curve on both sides of 
 the vertical part A 2 B { have to be equal. When K b is fixed the 
 right position of A 2 Bj. may be found by eye estimate. The eye 
 is rather sensitive for differences of small areas. Besides a shift 
 of A 2 Bi to the right or to the left enlarges one area and diminishes 
 the other so that even a slight deviation from the correct position 
 makes itself felt. In the same way the next step B 2 CiC 2 is 
 drawn with its vertical part B 2 C\ in such a position that the 
 areas on both sides are equal. The integral of the given curve 
 as far as K c will again have the same value as that of the stepping 
 line as far as K c . And so on for the other steps. The integral 
 of the stepping line is constructed in the way shown. It is 
 represented by a broken line beginning at the foot of the ordinate 
 of AI. The corners lie on the vertical parts of the steps or 
 their prolongations. It is readily seen that the broken line con- 
 sists of a series of tangents of the integral curve 
 
 y = 
 
 and that their points of contact with the integral curve lie on 
 the same verticals as the points AI, K b , K c , etc. (In Fig. 73 these 
 points are denoted 0, 2, 3, .) That these points lie on the 
 integral curve follows from the arrangement of the steps which 
 make the integral of the given function at K b , K c , equal to the 
 integral of the stepping line. Now in the points AI, K b , K c 
 the ordinates of the given curve coincide with those of the 
 stepping line. Hence both integral lines must for these abscissas 
 have the same direction. 
 1 In Fig. 73 the lower limit is 0.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 105 
 
 Having constructed the broken line and marked the points 
 2, 3, 4, (Fig. 73), the integral curve is drawn with a curved 
 ruler so as to touch the broken line in the points, 0, 2, 3, 
 As the given curve does not change its ordinate abruptly the 
 integral curve does not change its direction abruptly. The 
 drawing shows how well the integral curve is determined by the 
 broken line. There is practically no choice in drawing it any 
 other way without violating the conditions. 
 
 The ordinate of the integral curve is measured in the same 
 unit as the ordinate of the given curve y = f(x). It may some- 
 times be convenient to draw the ordinates of the integral curve 
 in a scale different from that of the ordinates of the given curve. 
 For instance the value of the integral may become so large that 
 measured in the same unit the ordinates of the integral curve 
 would pass the boundaries of the drawing board, or else they may 
 be so small that their changes cannot be measured with sufficient 
 accuracy. In the first case the scale is diminished, in the latter 
 case it is enlarged. This is done by altering the position of the 
 point P, the center of the pencil of rays that define the directions 
 of the broken line. If P approaches the directions Pa, P/3, 
 become steeper to the same degree as if keeping P unchanged we 
 had increased the ordinates of A\Ai, B\B%, in the inverse pro- 
 portion of the two distances PO. Hence by diminishing the 
 distance PO the ordinates of the resulting broken line are enlarged 
 in the inverse proportion. On the other hand, by increasing the 
 distance PO the ordinates of the resulting broken line are di- 
 minished in the inverse proportion of the distances, because the 
 change of the directions Pa, P/3, caused by a longer distance 
 PO is the same as if the ordinates of AiA 2 , BiB 2 , - were di- 
 minished in the inverse proportion. The broken line constructed 
 by means of the longer distance P'O will therefore be the same as 
 if the ordinates of the stepping line were diminished. It therefore 
 leads to an integral curve whose ordinates are diminished in the 
 same proportion (Fig. 74). 
 
 The graphical integration of
 
 106 
 
 GRAPHICAL METHODS. 
 
 Y = f(x)dz 
 
 is not limited to values x > a. The method is just as well applic- 
 able to the continuation of the integral curve for x < a. The 
 
 E, 
 
 FIG. 74. 
 
 H I 
 
 steps have only to be drawn from right to left. The lower limit 
 a determines the point where the integral curve intersects the 
 axis of x. 
 
 There is a method for the construction of the vertical parts 
 of the steps, which may in some cases be useful, though as a rule 
 we may dispense with it and fix their position by estimation. 
 
 Suppose that A and B (Fig. 75) 
 are two points where the curve is 
 intersected by the horizontal parts 
 of two consecutive steps and that 
 the curve between A and B is a 
 parabola whose axis is parallel to 
 the axis of x. The position of the 
 vertical part of the step between A 
 and B can be then found by a simple 
 
 construction. Through the center C of the chord AB (Fig. 75) 
 draw a parallel CD to the axis of x, D being the point of inter- 
 section with the parabola. The vertical part EH of the step in- 
 tersects CD in a point whose distance from C is twice the distance 
 
 FIG. 75.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 107 
 
 from D. That this is the right position of EH is shown as soon 
 as we can prove that the area AD EGA is equal to the rectangle 
 EHBG. The area ADGBA can be divided in two parts, the tri- 
 angle ABG and the part ADBCA between the curve and the 
 chord. The triangle is equal to the rectangle FIBG, whileADBCA 
 is equal to two thirds of the parallelogram MNBA, and hence 
 equal to the rectangle EH IF. Both together are therefore equal 
 to the rectangle EHBG, and the two areas between the stepping 
 line and the curve on both sides of EH are thus equal. 
 
 If the curve between A and B is sup- 
 posed to be a parabola with its axis par- 
 allel to the axis of ordinates the con- 
 struction has to be modified a little. 
 Through the center C of the chord AB 
 (Fig. 76) draw a vertical line CD as far 
 as the parabola. On CD find the point 
 K whose distance from C is double the 
 distance from D and draw through it a 
 parallel to the chord AB. This parallel 
 
 intersects a horizontal line through C at a point L. Then EH 
 must pass through L. This may be shown in the following way. 
 The area between the parabola ADB and the chord AB is equal 
 to two thirds of the parallelogram MNBA, MN being the tan- 
 gent to the parabola at the point D. If D' is the point of inter- 
 section of NN and the horizontal line through C, we have evi- 
 dently 
 
 CL = f CD'. 
 
 Therefore the rectangle EHIF is equal to the area ADB A be- 
 tween the parabola and the chord and EHBG is equal to ADGBA. 
 Any part of a curve can be approximated by the arc of a 
 parabola with sufficient accuracy if the part to be approximated 
 is sufficiently small. When the direction of the curve is nowhere 
 parallel to the axis of coordinates, both kinds of parabolas may 
 be used for approximation, those whose axes are parallel to the 
 axis of x and those whose axes are parallel to the axis of y. But 
 
 FIG. 78.
 
 108 
 
 GRAPHICAL METHODS. 
 
 when the direction in one of the points is horizontal (Fig. 76), 
 we can only use those with vertical axes and when the direction 
 in one of the points is vertical we can only use those with hori- 
 zontal axes. Accordingly we have to use either of the two con- 
 structions to find the position of the vertical part of the step. 
 Do not draw your steps too small. For, although the difference 
 between the broken line and the integral curve becomes smaller, 
 the drawing is liable to an accumulation of small errors owing 
 
 to the considerable number 
 of corners of the broken 
 line and little errors of 
 drawing committed at the 
 corners. Only practical ex- 
 perience enables one to find 
 the size best adapted to 
 the method. 
 
 Statical moments of areas 
 may be found by a double 
 
 graphical integration. Let us consider the area between the curve 
 y = /(*) (Fig. 77), the axis of x and the ordinates corresponding 
 to x = and x = . The statical moment with respect to the 
 vertical through x = is the integral of the products of each 
 element ydx and its distance x from the vertical 
 
 Fio. 77. 
 
 Let us regard M as a function of and differentiate it: 
 
 = + jf ydx. 
 
 That is to say, a graphical integration of the curve y = f(x~) 
 beginning at x = furnishes the curve whose ordinate is
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 109 
 
 Hence a second integration of this latter curve will furnish 
 the curve M as a function of . As M vanishes for = the 
 second integration must also begin at the abscissa x = 0. 
 
 , 
 
 FIG. 78. 
 
 Fig. 78 shows an example. Each ordinate of the curve found 
 by the second integration is the statical moment of the area on 
 the left side of it with respect to the vertical through this same 
 ordinate. The ordinate furthest to the right is the statical 
 moment of the whole area with respect to the vertical on the 
 right. The statical moment of the whole area with respect to a 
 vertical line through any point x\ is the integral 
 
 J^ (ari - x)ydx. 
 
 Considered as a function of xi its differential coefficient is 
 
 t J 
 
 K fe-. 
 
 That is to say, the differential coefficient is independent of Xi, 
 hence the statical moment is represented by a straight line. As 
 its differential coefficient is represented by a horizontal line 
 through the last point on the right of the curve 
 
 >*
 
 110 GRAPHICAL METHODS. 
 
 the direction of the straight line is found by drawing a line 
 through P and through the point of intersection Q of the hori- 
 zontal line and the axis of ordinates (Fig. 78). The position of 
 the straight line is then determined by the condition that 
 
 r 
 
 (xi - x)ydx 
 for 0-1= | is equal to the statical moment 
 
 - x}ydx. 
 
 We have therefore only to draw a parallel to PQ through the 
 last point R of the curve for M(i-) found by the second integration. 
 The ordinates of this straight line for any abscissa a'i represent 
 the values of 
 
 measured in the unit of length of the ordinates. The point of 
 intersection E with the axis of x determines the position of the 
 vertical in regard to which the statical moment is zero, that is to 
 say, the vertical through the center of gravity. 
 The moment of inertia of the area 
 
 about the axis x = is found in a similar way. It is exoressed 
 by the integral 
 
 Considered as a function of we find by differentiation 
 
 - x)ydx.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. Ill 
 
 That is to say, the differential coefficient is equal to double the 
 statical moment about the same axis. This holds for every value 
 of . Hence we obtain %T as a function of by integrating 
 the curve for 3f (). For = we have T = 0, so that the curve 
 begins on the axis of x at = 0, 
 The integral 
 
 '"yd* 
 
 is zero for x = a. The curve representing the integral has to 
 intersect the axis of x at x = a (admitting values of x > a and 
 x < a), and it is there that we begin the construction of the 
 broken line. If instead we begin it at the point x = a, y = c, 
 the only difference is that the whole integral curve is shifted 
 parallel to the axis of ordinates by an amount equal to c upwards 
 if c is positive, downwards if it is negative. But the form of the 
 curve remains the same. It is different when this curve is 
 integrated a second time. For instead of 
 
 f' 
 
 * ydx 
 
 we now integrate 
 
 ydx + c. 
 
 r 
 
 The ordinate of the integral curve is therefore changed by an 
 amount equal to c(x a) and besides if the second integral curve 
 is begun at x = a, y = c\ instead of x = a, y = the change 
 amounts to 
 
 c(x a) + ci, 
 
 so that the difference between the ordinates of the new integral 
 curve and the ordinates of the straight line 
 
 y = c(x a) + ci 
 
 is equal to the ordinates of the first integral curve (Fig. 79). 
 
 This effect of adding a linear function to the ordinates of the 
 integral curve is also attained by shifting the pole P upward or
 
 112 
 
 GRAPHICAL METHODS. 
 
 downward. For it evidently comes to the same thing whether 
 the curve to be integrated is shifted upward by the amount c or 
 whether the point P is moved downward by the same amount, so 
 that the relative position of P and the curve to be integrated 
 is the same as before. Changing the ordinate of P by c adds 
 
 ' fff(x)dxdx 
 ' "" 
 
 FIG. 79. 
 
 c(x a) to the ordinates of the integral curve. c(x a) is the 
 ordinate of a straight line parallel to the straight line from the 
 new position of P to the origin. 
 
 By this device of shifting the position of P upward or down- 
 ward the integral curve may sometimes be kept within the 
 boundaries of the drawing without any reduction of the scale of 
 ordinates. A good rule is to choose the ordinate of P about 
 equal to the mean ordinate of the curve to be integrated. The 
 ordinates of the integral curve will then be nearly the same at 
 both ends. The value of the integral 
 
 f 
 
 ydx 
 
 is equal to the difference between the ordinates of the integral 
 curve and the ordinates of a straight line parallel to PO through 
 the point of the integral curve whose abscissa is a.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 113 
 
 When the ordinate of P is accurately equal to the mean ordinate 
 of the curve to be integrated for the interval x = a to b the 
 ordinates of the integral curve will be accurately the same at the 
 two ends. But we do not know the mean ordinate before having 
 integrated the curve. 
 
 After having integrated we find the mean ordinate for the 
 interval x = a to 6 by drawing a straight line through P parallel 
 to the chord AB of the integral curve, A and B belonging to the 
 abscissas x=a and x=b. This 
 line intersects the axis of ordi- 
 nates at a point whose ordinate 
 is the mean ordinate. 
 
 Suppose a beam AB is sup- 
 ported at both ends and loaded 
 by a load distributed over the 
 
 beam as indicated by Fig. 80. That is to say, the load on dx is 
 measured by the area ydx. Let us integrate this curve graph- 
 ically, beginning at the point A with P on the line AB. The 
 final ordinate at B 
 
 f. 
 
 ydx 
 
 gives the whole load and is therefore equal to the sum of the two 
 reactions at A and B that equilibrate the load. Integrating this 
 curve again we obtain the curve whose ordinate is equal to 
 
 Y being written for 
 
 (* 
 
 ydx 
 
 J 
 
 The ordinate of this curve at any point x = represents the 
 statical moment of the load between the verticals x = a and 
 x = about the axis x = . Its final ordinate BM, Fig. 81, is 
 the moment of the whole load about the point B, and as the reac- 
 tions equilibrate the load it must be equal to the moment of the
 
 114 
 
 GRAPHICAL METHODS. 
 
 reactions about the same point and therefore opposite to the 
 moment of the reaction at A about B. If the reaction at A is 
 denoted by F a we therefore have 
 
 F a (b - a) = C Ydx. 
 
 That is to say, F a is equal to the mean ordinate of the curve 
 F = 
 
 in the interval x = a to 6. The mean ordinate is found by 
 drawing a parallel to AM through P which intersects the vertical 
 through A at the point F so that AF = F a . As DB is equal to 
 
 FIG. 81. 
 
 the sum of the two reactions a horizontal line through F will 
 divide BD into the two parts BG = F a and GD = F*. 
 
 Shifting the position of P to P' on the horizontal line FG 
 and repeating the integration 
 
 f 
 
 Jo, 
 
 Ydx, 
 
 we obtain a curve with equal ordinates at both ends. If we 
 begin at A it must end in B. Its ordinates are equal to the 
 difference between the ordinates of the chord AM and the curve 
 AM (Fig. 81), and represent the moment about any point of
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 115 
 
 the beam of all the forces on one side of the point (load and 
 reaction). 
 
 The area of a closed curve may be found by integrating over the 
 whole boundary. Suppose x = a and x = b t6 be the limits of 
 the abscissas of the closed curve, the vertical x = a touching the 
 curve at A and the vertical x = b at B (Fig. 82). By A and B 
 the closed curve is cut in two, both parts connecting A and B. 
 Let us denote the upper part by y = f\(x) and the lower part 
 by y = fz(x). The whole area is then equal to the difference 
 
 FIG. 82. 
 
 jT ftWdx - f*(x)dx, 
 
 or equal to 
 
 We begin the integral curve. over the upper part at the vertical 
 x = a at a point E, the ordinate of which is arbitrary, and draw 
 the broken line as far as F on the vertical x = b (Fig. 82). Then 
 we integrate back again over the lower part, continuing the 
 broken line from F to G. The line EG measured in the unit of 
 length set down for the ordinates is equal to the area measured 
 in units of area, this unit being a rectangle formed by PO and 
 the unit of ordinates. That is to say, the area is equal to the 
 area of a rectangle whose sides are PO and EG.
 
 116 
 
 GRAPHICAL METHODS. 
 
 The method is not limited to the case drawn in Fig. 82, where 
 the closed curve intersects any vertical not more than twice. A 
 more complicated case is shown in Fig. 83. But in all those cases 
 
 FIQ. 83. 
 
 where the object is not to find the integral curve but only to find 
 the value of the last ordinate the method, cannot claim to be 
 of much use, because it cannot compete with the planimeter. 
 
 FIQ. 84. 
 
 For the construction of the broken line we have drawn the 
 steps in such a manner that the areas on both sides of the vertical 
 part of a step between the curve and the stepping line are equal.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 117 
 
 It would have also been admissible to construct the stepping 
 line in such a way that the areas on both sides of the horizontal 
 part of a step are equal (Fig. 84). Only the broken line would 
 consist of a series of chords instead of a series of tangents of the 
 integral curve. The points K a , K b , , where the horizontal 
 parts of the steps intersect the curve would determine the ab- 
 scissas of the points of the integral curve, where its direction is 
 parallel to the direction of the broken line. But this forms very 
 little help for drawing the integral curve. That is the reason 
 why the former method where the broken line consists of a series 
 of tangents is to be preferred. However where the object is only 
 to find the last ordinate of the integral curve the two methods 
 are equivalent. 
 
 14. Graphical Differentiation. The graphical differentiation 
 of a function represented by a curve is not so satisfactory as the 
 graphical integration because 
 the values of the differential 
 coefficient are generally not 
 very well defined by the curve. 
 The operation consists in 
 drawing tangents to the given 
 curve and drawing parallels 
 through P to the tangents 
 (Fig. 85). The points of in- 
 tersection of these parallels 
 with the axis of ordinates fur- 
 nish the ordinates of the curve representing the derivative. 
 The abscissa to each ordinate coincides with the abscissa of the 
 point of contact of the corresponding tangent. The principal 
 difficulty is to draw the tangent correctly. As a rule it can be 
 recommended to draw a tangent of a given direction and then 
 mark its point of contact instead of trying to draw the tangent 
 for a given point of contact. A method of finding the point of 
 contact more accurately than by mere inspection consists in 
 drawing a number of chords parallel to the tangent and to 
 
 FIG. 85.
 
 118 
 
 GRAPHICAL METHODS. 
 
 bisect them. The points of bisection form a curve that inter- 
 sects the given curve at the point of contact (Fig. 86). When a 
 number of tangents are drawn, their points of contact marked 
 and the points representing the differential coefficient constructed, 
 
 the derivative curve has to be 
 drawn through these points. 
 This may be done more accur- 
 ately by means of the stepping 
 line. The horizontal parts of 
 the steps pass through the 
 points while the vertical parts 
 lie in the same vertical as the 
 point of intersection of two 
 
 consecutive tangents. The derivative curve connects the points 
 in such a way that the areas between it and the stepping line are 
 equal on both sides of the vertical parts of each step. Thus 
 the result of the graphical differentiation is exactly the same 
 
 Fia. 87. 
 
 figure that we get by integration, only the operations are carried 
 out in the inverse order. 
 
 A change of the distance PO (Fig. 87) changes the ordinates 
 of the derivative curve in the same proportion and for the same 
 reason that it changes the ordinates of the integral curve when we
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 119 
 
 are integrating, but in the inverse ratio. Any change of the or- 
 dinate of P only shifts the curve up or down by an equal amount, 
 so that if we at the same time change the axis of x and draw it 
 through the new position of P the ordinates of the curve will 
 remain the same and will represent the differential coefficient. 
 
 When a function f(x, y) of two variables is given by a diagram 
 showing the curves f(x, y) = const, for equidistant values of 
 f(x, y) the partial differential coefficients can be found at any 
 point X Q , 2/0 by means of drawing curves whose ordinates represent 
 f(x, yo) to the abscissa x orf(x Q , y) to the abscissa y and applying 
 the methods explained above. For this purpose a parallel is 
 drawn to the axis of x, for instance, through the point X Q , y 
 and at the points where it intersects the curves /(or, y) = const, 
 ordinates are erected representing the values of f(x, y ) in any 
 convenient scale. A smooth curve is then drawn though the 
 points so found and the tangent of the curve at the point XQ 
 furnishes the differential coefficient df/dx for x = XQ, y = y . 
 
 The differential coefficients df/dx, df/dy are best represented 
 graphically by a straight line starting from the point x, y to 
 which the differential coefficients correspond, and of such length 
 and direction that its orthogonal projections on the axis of x 
 and y are equal to df/dx and df/dy. This line represents the 
 gradient of the function f(x, y) at the point x, y. 1 It is normal 
 to the curve f(x, y) = const, that passes through the point x, y, 
 its direction being the direction of steepest ascent. Its length 
 measures the slope of the surface z = f(x, y) in the direction of 
 steepest ascent. This is shown by considering the slope in any 
 other direction. Let us change x and y by 
 
 r cos a, r sin a 
 and consider the corresponding change 
 
 Az = f(x + r cos a,y+r sin a) f(x, y) 
 of the function. By Taylor's theorem we can write it 
 1 See Chap. II, 10.
 
 120 GRAPHICAL METHODS. 
 
 r cos a + r sin a. + terms of higher order in r, 
 ox oy 
 
 a is the direction from the point x, y to the new point x -\- r cos a, 
 t/ + r sin a and r is the distance of the two points. Dividing 
 Az by r and letting r approach to zero we find 
 
 AS a/ . a/ . 
 
 hm = cos a + sin a. 
 r dx By 
 
 This expression measures the slope of the surface z = f(xy] 
 in the direction a. Now let us introduce the length / and the 
 angle X of the gradient, and write 
 
 ~ = I cos X. ~ = I sin X. 
 dx dy 
 
 Then we have 
 
 cos a + sin a = I cos (a X). 
 dx dy 
 
 That is to say, the slope in any direction a is proportional to 
 cos (a X), it is a maximum in the direction of the gradient 
 (a = X) and zero in a direction perpendicular to it and negative 
 in all directions that form an obtuse angle with it. When all 
 three coordinates are measured in the same unit, the length of 
 I measured in this unft is equal to the tangent of the angle of 
 steepest ascent. Hence the length of the gradient varies with 
 the unit of length. When the unit of length in which the values 
 of f(xy) are plotted is kept unaltered, while we change the unit 
 of length corresponding to the values x and y, the length of the 
 gradient varies with the square of the unit of length. 
 
 15. Differential Equations of the First Order. In the problem 
 of solving a differential equation of the first order 
 
 by graphical methods the first question is how to represent 
 the differential equation graphically. If x and y are meant to 
 be the values of rectangular coordinates, the geometrical meaning
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 121 
 
 of the differential equation is that at every point x, y, where 
 f(x, y} is defined, the equation prescribes a certain direction for 
 the curve that satisfies it. Let us suppose curves drawn through 
 all those points for which f(x, y) has certain constant values. 
 Each curve then corresponds to a certain direction or the opposite 
 direction. Let us distinguish the curves by different numbers or 
 letters and let us draw a pencil of rays together with the curves 
 and mark the rays with the same numbers or letters in such a way 
 that each of them shows the direction corresponding to the 
 
 FIG. 88. 
 
 curve marked with that particular number or letter (Fig. 88). 
 Our drawing of course only comprises a certain region in which 
 we propose to find the curves satisfying the differential equation. 
 It may be that f(xy) is defined beyond the boundaries of our 
 drawing. Those regions have to be dealt with separately. 
 
 The graphical representation of the differential equation in 
 the region considered consists in the correspondence between 
 the curves and the rays. It is important to observe that this 
 representation is independent of the system of coordinates by 
 means of which we have deduced the curves from the equation
 
 122 GRAPHICAL METHODS. 
 
 We can now introduce any system of coordinates , 77 and find 
 from our drawing the equation 
 
 that is to say, we can find the value of <p(%, 77) at any point , 77 
 of our drawing. If, for instance, the unit of length is the same 
 for and 77 we draw a line through the center of the pencil of rays 
 in the direction of the positive axis of and a line perpendicular to 
 it at the distance 1 from the center. The segment on the second 
 line between the first line and the point of intersection with one 
 of the rays measured in units of length and counted positive in 
 the direction of positive 77 furnishes the value of ?>(, 77) for all 
 the points , 77 corresponding to that particular ray. In this 
 respect the graphical representation of a differential equation 
 is superior to the analytical form, in which certain coordinates 
 are used and the transformation to another system of coordinates 
 requires a certain amount of calculation. 
 
 Now let us try to find the curve through a given point P on 
 the curve marked (a) (Fig. 88) that satisfies the differential equa- 
 tion. We begin by drawing a series of tangents of a curve 
 that is meant to be a first approximation. Through P we draw 
 a parallel to the ray (a) as far as the point Q somewhere in the 
 middle between the curves (a) and (6). Through Q we draw a 
 parallel to the ray (6) as far as R somewhere in the middle 
 between the curves (&) and (c). Through R we again draw a 
 parallel to the ray (c) and so on. The curve touching this 
 broken line at the points of intersection with the curves (a), 
 (6), is a first approximation. But we need not draw this 
 curve. In order to find a better approximation we introduce a 
 rectangular system of coordinates x, y, laying the axis of x some- 
 what in the mean direction of the broken line. Let us denote 
 by i/i the function of x that corresponds to the curve forming the 
 first approximation. The second approximation y z is then ob- 
 tained as an integral curve of f(x, y\}, that is, of dyi/dx
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 123 
 
 I 
 
 J x 
 
 denoting by x p , y p , the coordinates of P. For this purpose the 
 curve whose ordi nates are equal to f(x, yi) or dyi/dx has to be con- 
 structed first. The values of f(x, yi) are found immediately at 
 the points where the first approximation intersects the curve 
 (a), (6) by differentiation in the way described above. A 
 line is drawn through the center of the pencil of rays parallel to 
 the axis of x and a line perpendicular to it at a convenient dis- 
 tance from the center. This distance is chosen as the unit of 
 length. The points of intersection of this line with the rays de- 
 termine segments whose lengths are equal to the values of f(x, yi) 
 on the corresponding curves. These values are plotted as ordi- 
 nates to the abscissas of the points where the first approximation 
 intersects the curves (a), (6), and a curve 
 
 is drawn (Fig. 88). This curve is integrated graphically begin- 
 ning at the point P and the integral curve is a second approxi- 
 mation. Again we need not draw the curve. The broken line 
 suffices, if we intend to construct a third approximation. In 
 this case we have to repeat the foregoing operation. This can 
 now be performed much quicker than in the first case because the 
 values of f(x, y) on the curves (a), (6), have already been 
 constructed and are at our disposal. In order to find the curve 
 
 we have only to shift the same ordinates to new abscissas and 
 make these coincide with the abscissas of the points where the 
 second approximation intersects the curves (a), (6), . The 
 curve 
 
 is then drawn and integrated graphically, beginning at the point 
 P.
 
 124 GRAPHICAL METHODS. 
 
 Suppose now the integral curve did not differ from the second 
 approximation, it would mean that 
 
 2/2 = y P + J f(x,y 2 )dx, 
 or that 
 
 that is to say, that y z satisfies the differential equation. 
 
 If there is a perceptible difference the integral curve represents 
 a third approximation. It has been shown by Picard that pro- 
 ceeding in this way we find the approximations (under a certain 
 condition to be discussed presently) converging to the true solu- 
 tion of the differential equation, so that after a certain number 
 of operations the error of the approximation must become 
 imperceptible. 
 
 Denoting by y n the function of the nth approximation we have 
 
 2/n+i = y P + I f(x, yn)dx. 
 
 y. 
 
 The true solution w r ith the same initial conditions y = y p for 
 x = Xp satisfies the equation 
 
 c x 
 
 y = yp + I /fo y}- 
 
 Hence 
 
 y n+1 - y = J [f(x, y n ) - f(x, y)]dx, 
 or 
 
 Let us now suppose that the absolute value of 
 
 f(x, y n ) f(x, y) 
 
 yn-y 
 
 for all the values of x, y, y n within the considered region does
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 125 
 
 not surpass a certain limit M, then it follows that a certain relation 
 must exist between the maximum error of y n , which we denote by 
 e n and the maximum error of y n +i, which we denote by e n +i* 
 The absolute value of the integral not being larger than 
 
 ( | x x n | denoting the absolute value of x x n } we have 
 e n +i ^ M | x x n | e n . 
 
 Hence as long as the distance x x n over which the integration 
 is performed is so small that 
 
 M | X - X n | ^ k < 1, 
 
 k being a constant smaller than one, the error of y n+ i cannot be 
 larger than a certain fraction of the maximum error of y n . 
 But in the same way it follows that the error of y n cannot be 
 larger than the same fraction of the maximum error of y n -i, and 
 
 so on, so that 
 
 e n+ i ke n k z e n -i kei. 
 
 But as ei is a constant and k a constant smaller than one, k n ei 
 must be as small as we please for a sufficient large value of n. 
 That is to say, the approximations converge to the true solution. 
 M being a given constant the condition of convergence 
 
 M | x - x p | < k < 1 
 
 limits the extent of our integration in the direction of the axis of x. 
 But it does not limit our progress. From any point P r that we 
 have reached with sufficient accuracy we can make a fresh start, 
 choosing a new axis of x suited to the new situation. As a 
 rule it does not pay to trouble about the value of M and to try 
 to find the extent of the convergence by the help of this value. 
 The actual construction of the approximations will show clearly 
 enough how far to extend the integration. As far as two consecu- 
 tive approximations show no difference they represent the true 
 curve.
 
 126 GRAPHICAL METHODS. 
 
 Suppose that 
 
 has the same sign for all values x, y, y n concerned. Say it is 
 negative. Suppose further that y n y is of the same sign for 
 the whole extent of the integration 
 
 that is to say, the approximative curve y n is all on one side of the 
 true curve. Then if x x p is positive, y n n ~ y must evidently 
 be of the opposite sign from y n y, or the approximative curve 
 y n+ i is all on the other side of the true curve from y n . For these 
 and all following approximations the true curve must lie between 
 two consecutive approximations. If the first approximation y is 
 all on one side of the true curve the theorem holds for any two 
 consecutive approximations. This is very convenient for the esti- 
 mation of the error. 
 In Fig. 88 
 
 y - y 
 
 is negative from the point P as far as somewhere near S. The 
 first approximation is all on the upper side of the true curve. 
 Therefore the second approximation must be below the true 
 curve at least as far as somewhere near S. 
 
 When the sign is positive the same theorem holds for negative 
 values of x x p . If the integration has been performed in the 
 positive direction of x, it may be a good plan to check the result 
 by integrating backwards, starting from a point that has been 
 reached and to try if the curve gets back to the first starting 
 point. In this direction we profit from the advantage of the 
 true curve lying between consecutive approximations and are 
 better able to estimate the accuracy of our drawing. 
 
 We have seen that the convergence depends on the maximum
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 127 
 
 absolute value of 
 
 f(x, y n ) f(x, y) 
 
 y n - y 
 
 for all values of x, y, y n concerned. In order to find the maximum 
 value we may as well consider 
 
 for all values of x, y within the region considered. For if we 
 assume df/dy to be a continuous function of y, it follows that 
 the quotient of differences 
 
 y n - y 
 
 must be equal to df/dy taken for the same value of x and a value 
 of y between y and y n . This is immediately seen by plotting 
 f(x, y) as ordinate to the abscissa y for a fixed value of x. The 
 value of the quotient of differences is determined by the slope 
 of the chord between the two points of abscissas y and y n . The 
 slope of the chord is equal to the slope of the curve at a certain 
 point between the ends of the chord. The value of df/dy at this 
 point is equal to the value of 
 
 yn y 
 
 Now let us consider how the coordinate system may be chosen 
 in order to make df/dy as small as possible and thus obtain the 
 best convergence. For this purpose let us investigate how the 
 value of df/dy changes at a certain point, when the system of 
 coordinates is changed. 
 
 Let us start with a given system of rectangular coordinates , 
 77 with which the differential equation is written 
 
 The direction of the curve satisfying the differential equation
 
 128 GRAPHICAL METHODS. 
 
 forms a certain angle a with the positive axis of determined by 
 
 tg a = f$ = ^' ^ 
 
 (assuming the coordinates to be measured in the same unit). 
 Now let us introduce a new system of rectangular coordinates 
 x, y connected with the system , rj by the equations 
 
 x = % cos co + TJ sin co, 
 y = sin co -f- T/ cos co, 
 which are equivalent to 
 
 = x cos co y sin co, 
 77 = x sin co + y cos cc, 
 
 co being the angle between the positive direction of x and the 
 positive direction of , counted from towards x in the usual way. 
 The angle formed by the direction of the curve with the positive 
 direction of the axis of x is a co, and therefore 
 
 g=tg (a -co) =/(*,</). 
 
 Consequently we obtain for a given value of w 
 
 df = 1 da 
 
 dy ~ cos 2 (a co) dy ' 
 
 or remembering that a is given as a function of and 17, 
 
 df 1 ( da . , da \ 
 
 = IT, N ' I sin co -f- cos co I. 
 
 dy cos 2 (a co) V d &l ) 
 
 For simplicity's sake we shall assume that the axis of is the 
 tangent of the curve <p(,i\) = const, that passes through the 
 given point, so that dafd!- = 0. 
 We then have 
 
 df 1 da 
 
 = r7 : T- cos co, 
 
 dy cos j (a co) drj 
 
 and our object is to find how df/dy varies for different values of
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 129 
 
 co. The value of da/dy is independent of co; it denotes the value 
 of the gradient of a, which we represent by a straight line drawn 
 from the origin A (Fig. 89) perpendicular to the curve a = const, 
 or <f>(, 17) = const. 
 
 It is no restriction to assume the value of da/drj positive; it 
 only means that the direction of the positive axis of t] is chosen 
 
 FIG. 89. 
 
 in the direction of the gradient. Let us draw the line AB (Fig. 
 89) in the direction of the positive axis of and of the same length 
 as the gradient. 
 
 In order to show the values of df/dy for the different positions 
 of the axis of x let us lay off the value of dffdy as an abscissa. 
 For instance for co = or, df/dy assumes the value 
 
 da 
 
 cos a. 
 dr, 
 
 The abscissa corresponding to this value is AB' (Fig. 
 10 
 
 ), the
 
 130 
 
 GRAPHICAL METHODS. 
 
 orthogonal projection of AB on the axis of x. For any other 
 position AC (Fig. 89) corresponding to some other value of to, 
 we find da/dri cos to by orthogonal projection of AB on AC. Then 
 the division by cos (a to) furnishes AC' and a second division 
 by cos (a to) leads to AC. Thus a certain curve can be 
 constructed whose polar coordinates are r = df/dy and to, the 
 equation in polar coordinates being 
 
 da cos to .. da 
 
 r 5-7 r- or [r cos (or to)l z = r cos to. 
 
 drj COS 2 (a to) drj 
 
 In rectangular coordinates , y the equation assumes the form 
 (cos a + sin arj) 2 = . 
 
 This shows that the equation is a parabola, the axis of which is 
 perpendicular to the direction a. AB' is a chord and the gradient 
 
 FIG. 90. 
 
 AG is a tangent of the parabola. Bisecting AB' in E, drawing EK 
 perpendicular to AB' as far as the axis of 77 and bisecting EK in 
 D, we find D the apex of the parabola. The three points A,B',D 
 together with the gradient will suffice to give us an idea of the 
 size and sign of df/dy for the different positions of the positive 
 axis of x.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 131 
 
 df/dy vanishes when the axis of x is perpendicular to the curve 
 a = const., so that it seems as if this were the most favorable 
 position. We must, however, bear in mind that the axis of x 
 is kept unaltered for a certain interval of integration. When we 
 pass on to other points the axis of x is no longer perpendicular 
 to the curve a = const, there. The position of the axis of x is 
 good when the average value of df/dy is small. In Fig. 90 the 
 parabolas are constructed for a number of points on the first 
 approximation of a curve satisfying the differential equation. 
 
 If we want to make use of the parabolas to give us the numerical 
 values of df/dy the unit of length must also be marked in which 
 the coordinates are measured. The numerical value of df/dy 
 varies as the unit of length and therefore the length of the line 
 representing it must vary as the square of the unit of length. 
 But if we draw a line whose length measured in the same unit is 
 
 equal to -TTTT > this line would be independent of the unit of 
 length. For if I is the line representing the unit of length and 
 I', I" the lines representing the values df/dy and 
 
 would be the ratio l'/l and T- the ratio I" /I; hence I" = P/l'. 
 
 Since I' varies as P with the change of the unit of length I" is 
 independent of the unit of length. This line I" represents the 
 limit beyond which the product 
 
 becomes greater than 1. If df/dy remained the same this would 
 mean the limit beyond which the convergence of the process of 
 
 approximation ceases. We might lay off the length of TTT- in 
 
 the different directions in the same way as df/dy has been laid 
 off. The result is a curve corresponding, point by point, to the 
 parabola, the image of the parabola according to the relation of 
 reciprocal radii. But all these preparations .as a rule would not
 
 132 GRAPHICAL METHODS. 
 
 pay. It is better to attack the integration at once with an axis 
 of x somewhat perpendicular to the curves a = const, as long 
 as the direction of the curve forms a considerable angle with 
 the curve a = const, and to lose no time in troubling about the 
 very best position. The convergence w r ill show itself, w T hen the 
 operations are carried out. When the angle between the direction 
 of the curve that satisfies the differential equation and the curve 
 a = const, becomes small the apex of the parabola moves far 
 away and when the direction coincides with that of the curve 
 a = const, the parabola degenerates into two parallel lines per- 
 pendicular to the direction of the curve a. = const. In this case 
 the best position for the axis of x is in the direction of the curve 
 a const. Without going into any detailed investigation about 
 the best position of the axis of x we can establish the general rule 
 not to make the axis of x perpendicular to the direction of the 
 curve satisfying the differential equation, that is to say, not to 
 make it parallel to the axis of the parabola. But we hardly need 
 pronounce this rule. In practice it would enforce its own observ- 
 ance, because for that position of the axis of x not only df/dy but 
 also f(x, y) are infinite and it would become impossible to plot 
 the curve Y = f(x, y\). 
 
 There is another graphical method of integrating a differential 
 equation of the first order 
 
 which in some cases may well compete with the first method. 
 Like the first it is the analogue of a certain numerical method. 
 The numerical method starts from given values x, y and cal- 
 culates the change of y corresponding to a certain small change 
 of x. Let h be the change of x and k the change of y, so that 
 x + h, y + k are the coordinates of a point on the curve satisfying 
 the differential equation and passing through the point x, y. k is 
 calculated in the following manner. We calculate in succession 
 four values ki, k 2 , k 3 , & 4 by the following equations
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 133 
 
 fa = f(z, y)h, 
 
 fa = 
 
 fa = flx + 2>y + ^jh, 
 fa=f(x+h,y+fa)h. 
 We then form the arithmetical means 
 
 fa fa + fa 
 
 and *"~~~ 
 
 and find with a high degree of approximation as long as h is 
 not too large 
 
 k = p+$(q- p). 1 
 
 The new values 
 
 X = x+h, Y=y + k 
 
 are then substituted for x and y and in the same way the coordi- 
 nates of a third point are calculated and so on. 
 
 This calculation may be performed graphically in a profitable 
 manner, if the function f(x, y) is represented in a way suited to 
 
 FIG. 91. 
 1 See W. Kutta, Zeitschrift fur Malhematik und Physik, Vol. 46, p. 443.
 
 134 GRAPHICAL METHODS. 
 
 the purpose. Let us suppose a number of equidistant parallels 
 to the axis of ordinates : x = XQ, x = x\, x = x 2 , x = x%, 
 Along these lines f(x, y) is a function of y. Let us lay off the 
 values of f(x, y) as ordinates to the abscissa y, the axis of y being 
 taken as the axis of abscissas. We thus obtain a number of 
 curves representing the functions f(x , y), /(ar lf y), f(xz, y), 
 Starting from a point A(x , y Q } on the first vertical x = X Q (Fig. 
 91) we proceed to a point B : on the vertical x = x 2 in the following 
 way. By drawing a horizontal line through A we find the 
 point A' on the curve representing f(x , y). Its ordinate is equal 
 to f(x , i/o). Projecting the point A' onto the axis of x we find A" 
 and draw the line PA". P is a point on the negative side of 
 the 2/-axis and PO is equal to the unit of length by which the 
 lines representing f(x, y) are measured. Thus 
 
 OA"IPO = f(x Q , 2/ ). 
 
 Now we draw ABi perpendicular to PA", so that if h and /,'i 
 denote the differences of the coordinates of A and B, we have 
 
 kjh = OA"/PO, 
 h = f(x Q , yo)h. 
 
 From Ci the point of intersection of the line AE\ and the vertical 
 x = x\ we find C\ and C\" in the same way as we found A' and A" 
 from A, only that C\ is taken in the curve representing the 
 values of f(xi, y}, and draw the line AB 2 perpendicular to PC\". 
 Denoting the difference of the ordinates of A and B 2 by A- 2 we have 
 
 h Od" 
 
 h = -po = 
 
 or 
 
 From Cz the point of intersection of the line AB% and the 
 vertical x = x\ we find in the same way a point B$ on the vertical
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 135 
 
 x = x 2 and the difference & 3 between the ordinate of B 3 and that 
 of A is 
 
 From j? 3 we pass horizontally to B 3 ' on the curve representing 
 f(x 2 y) and vertically down to J5 3 ". The line AB is then drawn 
 perpendicular to PB 3 ", so that the difference & 4 between the 
 ordinates of B^ and A is 
 
 The bisection of #2^3 and of -Z?il?4 gives us the points E\ and 2 
 and the point B is taken between EI and 2, so that its distance 
 from "1 is half its distance from 2 . The point B is with a high 
 degree of approximation a point of the curve that passes through 
 A and satisfies the differential equation. 
 
 B is then taken as a new point of departure instead of A, and 
 in this manner a series of points of the curve are found. 
 
 In order to get an idea of the accuracy attained the distance 
 of the vertical lines is altered. For instance, we may leave out 
 the verticals x = x\ and x = x 3 , and reach the point on the 
 vertical x = x^ in one step instead of two. The error of this 
 point should then be about sixteen times as large as the error on 
 the same vertical reached by two steps, so that the error of the 
 latter should be about one-fifteenth of the distance of the two. 
 If their distance is not appreciable the smaller steps are evidently 
 unnecessarily small. 
 
 The values of f(x, y) may become so large that an incon- 
 veniently small unit of length must be applied to plot them. In 
 this case x and y have to change parts and the differential 
 equation is written in the form 
 
 dx 1 
 
 dy f(x, y) ' 
 The values of l/f(x, y) are then plotted for equidistant values of
 
 136 GRAPHICAL METHODS. 
 
 y as ordinates to the abscissa x and the constructions are changed 
 accordingly. 
 
 16. Differential Equations of the Second and Higher Orders. 
 Differential equations of the second order may be written in the 
 form 
 
 'dx' 
 
 Let us introduce the radius of curvature instead of the second 
 differential coefficient. Suppose we pass along a curve that 
 satisfies the equation and the direction of our motion is deter- 
 mined by the angle a it forms with the positive axis of x (counted 
 in the usual way from the positive axis of x through ninety 
 degrees to the positive axis of y and so on), s being the length of 
 the curve counted from a certain point from which we start. 
 We then have 
 
 dy dx 
 
 y- = tg a, -j- COS a. 
 dx ds 
 
 Consequently 
 
 d 2 y _ 1 da 1 da 
 
 dx 2 cos 2 a dx cos 3 a ds ' 
 or 
 
 da d*y 
 
 T -** 
 
 da/ds measures the "curvature," the rate of change of direction 
 as we pass along the curve, counted positive when the change 
 takes place to the side of greater values of a (if the positive axis 
 of x is drawn to the right and the positive axis of y upwards a 
 positive value of da/ds means that the path turns to the left). 
 Let us count the radius of curvature with the same sign as da/ds 
 and let us denote it by p. Then we have 
 
 - = cos 3 a/(z, y,tg). 
 
 Thus the differential equation of the second order may be said 
 to give the radius of curvature as a function of x, y, a, that is to 
 say, as a function of place and direction.
 
 DIFFERENTIAL AND IXTEGEAL CALCULUS. 137 
 
 Let us assume that this function of three variables is repre- 
 sented by a diagram, so that the length and sign of p may quickly 
 be obtained for any point and any direction. 
 
 Starting from any given point in any given direction we can 
 then approximate the curve satisfying the differential equation 
 by a series of circular arcs. Let A (Fig. 92) be the starting point. 
 We make M a A perpendicular to the given direction and equal to 
 p in length. For positive values of p, M a must be on the positive 
 side of the given direction, for negative 
 values on the negative side. M a is 
 the center of curvature for the curve 
 at A. With M a as center and M a A 
 as radius we draw a circular arc AB 
 and draw the line BM a . On this line 
 or on its production we mark the . 
 point M b at a distance from B equal 
 to the value of p that corresponds to 
 B and to the direction in which the p^,. 92 . 
 
 circular arc reaches B. With 3/ b as 
 center and MiB as radius we draw a circular arc BC and so on. 
 
 The true curve changes its radius of curvature continuously, 
 while our approximation changes it abruptly at the points 
 A, B, C, - - . The smaller the circular arcs the less will accu- 
 rately-drawn circular arcs deviate from the curve. But it must 
 be kept in mind that small errors cannot be avoided, when 
 passing from one arc to the next. Hence, if the arcs are taken 
 very small so that their number for a given length of curve 
 increases unduly, the accuracy will not be greater than with 
 somewhat longer arcs. The best length cannot well be defined 
 mathematically; it must be left to the experience of the draughts- 
 man. 
 
 Some advantage may be gained by letting the centers and the 
 radii of the circular arcs deviate from the stated values. The 
 circular arc AB (Fig. 92) is evidently drawn with too small a 
 radius because the radius of the curve increases towards B. If
 
 138 GRAPHICAL METHODS. 
 
 we had taken the radius equal to M b B it would have been too 
 large. A better approximation is evidently obtained by making 
 the radius of the first circular arc equal to the mean of M a A and 
 M b B, and the direction with which it reaches B will also be closer 
 to the right direction. 
 
 To facilitate the plotting an instrument may be used consisting 
 of a flat ruler with a hole on one end for a pencil or a capillary 
 tube or any other device for tracing a line. A straight line 
 with a scale is marked along the middle of the ruler and a little 
 tripod of sewing needles is placed with one foot on the line and 
 two feet on the paper. Thus the pencil traces a circular arc. 
 When the radius is changed, the ruler is held in its position by 
 pressing it against the paper until the tripod is moved to a new 
 position. By this device the pencil must continue its path in 
 exactly the same direction, while with the use of ordinary com- 
 passes it is not easy to avoid a slight break in the curve at the 
 joint of two circular arcs. 
 
 Another method consists in a generalization of the method 
 for the graphical solution of a differential equation of the first 
 order. 
 
 A differential equation of the second order 
 
 may be written in the form of two simultaneous equations of the 
 first order: 
 
 dy 
 
 _ ^L ty 
 
 dx~ Z ' 
 
 Let us consider the more general form, in which the differential 
 coefficients of two functions y, z of x are given as functions of 
 x, y, z: 
 
 I -MM.
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 139 
 
 ~ = g(x,y,z). 
 
 We may interpret x, y, z as the coordinates of a point in space 
 and the differential equation as a law establishing a certain 
 direction or the opposite at every point in space where f(x, y, 2) 
 and g(x, y, 2) are defined. A curve in space satisfies the dif- 
 ferential equation, when it never deviates from the prescribed 
 direction. Its projection in the xy plane represents the function 
 y and its projection in the xz plane represents the function z. 
 
 Let us represent y and z as ordinates and x as abscissa in the 
 same plane with the same system of coordinates. Any point in 
 
 FIG. 
 
 space is represented by two points with the same abscissa. The 
 functions f(x, y, z) and g(x, y, z) we suppose to be given either 
 by diagrams or by certain methods of construction or calculation. 
 For any point that we have to deal with, the values of f(x, y, 2) 
 and g(x, y, 2) are plotted as ordinates to the abscissa x, but for 
 clearness sake not in the same system of coordinates as y and 2, 
 but in another system with the same axis of ordinates and an 
 axis of x parallel to the first and removed far enough so that the 
 drawings in the two systems do not interfere with one another.
 
 140 GRAPHICAL METHODS. 
 
 Starting from a certain point P(x p , y p , z p ) in space we represent 
 it by the two points PI(X P , y p } and PZ(X P , z p ) in the first system 
 and the values of f(x p , y p , z p ) and g(x p , y p , z p ) by the two points 
 AI and AZ in the second system of coordinates (Fig. 93). The 
 points AI and AZ determine certain directions MA\, and MAz' 
 of the curves x, y and x, z, the point M (Fig. 93) being placed at a 
 distance from the axis of ordinates equal to the unit of length by 
 which the ordinates representing f(x, y, z) and g(x, y, z) are 
 measured. Through PI and P 2 we draw parallels to MAi and 
 MAz as far as Qi and Qz with the coordinates x q , y q and x q , z q . 
 With these coordinates the values f(x q , y q , z q ) and g(x q , y q , z q ) 
 are determined, which we represent by the ordinates of the 
 points BI, Bz. These points again determine certain directions 
 parallel to which the lines QiRi and QzRz are drawn, etc. In this 
 manner we find first approximations y\ and Zi for the functions 
 y and z and corresponding to these approximations we find 
 curves representing f(x, yi, Zi) and g(x, y\, Zi). These curves are 
 now integrated graphically, the integral curve of f(x, yi, zO 
 beginning at PI and the integral curve of g(x, y\, Zi) at P 2 and 
 lead to second approximations y z and ZQ ' 
 
 = y P + I f(x, yi, zjdx 
 J *P 
 
 = z p + I g(x, yi* *i)d 
 " 
 
 For these second approximations the values of f(x, y^, Za) and 
 g(x, y^, %z) are determined at a number of points along the curves 
 x, 7/2 and x, Zz sufficiently close to construct the curves representing 
 f(x, yz, Zi) and g(x, y 2 , Zz). By their integration a third approxi- 
 mation 2/3, Zs is obtained 
 
 2/3 = y P + I f(x, yz, *2)dx, 
 JX P 
 
 = z p + I g(x, yz, zz)dx, 
 
 r
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 141 
 
 and so on as long as a deviation of an approximation from the 
 one before can still be detected. As soon as there is no deviation 
 for a certain distance x x p the curve represents the true solu- 
 tion (as far as the accuracy of the drawing goes). The curve is 
 continued by taking its last point as a new starting point for a 
 similar operation. 
 
 The distance over which the integral is taken can in general 
 not surpass a certain limit where the convergence of the approxi- 
 mations ceases. But we are free to make it as small as we please 
 and accordingly increase the number of operations to reach a 
 given distance. It is evidently not economical to make it too 
 small. On the contrary, we shall choose it as large as possible 
 without unduly increasing the number of approximations. 
 
 In the case of a differential equation 
 
 j?-f (**:, 
 
 we have f(x, y, z) = z, and the curve z, x is identical with the 
 curve representing the values of f(x, y, z). We shall therefore 
 draw it only once. 
 
 The proof of the convergence of the approximations is almost 
 the same as in the case of the differential equation of the first 
 order. 
 
 For the n + 1 st approximation we have 
 
 y n+l = y p + J f(x, y n , z n )dx; z n+ i = z p + J g(x, y n , z n }dx. 
 
 For the true curve that passes through the point x p , y p , z p we 
 find by integration 
 
 y 
 
 hence 
 
 r 
 f(x, y, z)dx\ z = z p + I g(x, y, z)dx; 
 Jr f 
 
 y n +i - y = I [f(x, Vn, z) - f(x, y, z)]dx; 
 
 " x p' 
 
 z n+ i - z = J [g(x, y n , z n ) - g(x, y, z)]dx.
 
 142 GRAPHICAL METHODS. 
 
 Now let us write 
 
 -, , f( s f(x - 
 
 /(*, y n , z n } - f(x, y, z) = - 
 
 yn y 
 
 f(x, y, z n ) - f(x, y, z) _ 
 
 z n z 
 and similarly 
 
 g(x, y n , z n ) </(#, y, z n ) 
 
 yn y 
 
 g(x,y,z n }-g(x,y,z) 
 
 Zn-z ~ (Zn ~ 2) * 
 The quotients of differences 
 
 /fcy,g.)-/(*,y,g.) 
 
 and the three others are equal to certain values of df/dy, df/dz, 
 dg/dy, dg/dz for values of y, z between y and y n and between z 
 and z n (y, y n , z, z n not excluded). Let us assume that for the 
 region of all the values of x, y, z concerned the absolute value of 
 df/dy and dffdz, is not greater than MI, and that of dg/dy and 
 dg/dz not greater than 3/ 2 , and that d n , e n denote the maximum 
 of the absolute values of y y n and z z n in the interval 
 x p to x. Then it follows that the absolute values of 
 
 f(x, y n , z n ) f(x, y, z) and g(x, y n , z n ) g(x, y, z) 
 are not greater than 
 
 Hence for the maximum values of y n+i y and z n+ i z, which 
 are denoted by 5 n+1 and e^+i we obtain the limits 
 
 and 
 
 8 n +i + e n+ i ^ (3/i + 3/ 2 ) | x x p (8 n + c n ). 
 
 If therefore the interval x x p of the integration is so far 
 reduced that 
 
 (3/i + 3/ 2 ) | x x p | ^ k < 1,
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 143 
 
 5 n+ i + e n+ i is not larger than the fraction k of (8 n + e n ), but 
 from the same reason 
 
 (5 B + e n ) ^ ^(Sn-! + _!), (5 n _! + e n _ t ) < &(5 n _ 2 + 6^2), etc.; 
 
 therefore 
 
 5 n+1 + e n+1 ^ &"(5i + ei). 
 
 That is to say, for a sufficiently large value of n 8n+i and e n +i 
 will both become as small as we please. 
 
 As in the case of the differential equation of the first order it is 
 not worth while, as a rule, to investigate the convergence for the 
 purpose of finding a sufficiently close approximation by graphical 
 methods. It is better at once to tackle the task of drawing the 
 approximations and to repeat the operations until no further 
 improvement is obtained. The curve will then satisfy the 
 differential equation as far as the graphical methods allow it 
 to be recognized. 
 
 ^Yhen the values of f(x, y, z) or g(x, y, z) become too large 
 we can have recourse to the same device that we found useful 
 with the differential equation of the first order. Instead of x, 
 one of the other two variables y or z may be considered as inde- 
 pendent, so that the equations take the form 
 
 dx _ 1 dz _ g(x, y, z) 
 
 dy ~ f(x, y, z)' dy~ f(x, y, z) ' 
 or 
 
 dx _ I dy _ f(x, y, z) 
 
 dz ~ g(x, y, z) ' dz g(x, y, z) ' 
 
 or we may introduce a new system of coordinates x f , y', z' and 
 consider the resulting differential equations. 
 
 The second method for the integration of differential equations 
 of the first order can also be generalized to include the second 
 order. Let us again consider the more general case 
 
 Starting from a point x, y, z the changes of y and z (denoted by
 
 144 GRAPHICAL METHODS. 
 
 k and /) can be calculated for a small change A of z by the fol- 
 lowing formulas analogous to those used for one differential 
 equation of the first order: 
 
 ki = /(#, y, z)A; l\ = g(x, y, z)h; 
 
 fc = /(*+ \ , y+ | , ^+ |) A; fe = ^ (.r+ f~, y+ 1 , 2 + |)A; 
 
 :> 
 
 *4 = /(a: + A, y+ 3, 2 + ^s)A; k = flf(ar + A, y + h, z + I 3 )h; 
 
 h + h k, + h , k + k h + k 
 
 P = 2 ' q = 2 ; P = 2 ' q = 2 ' 
 
 and with a high degree of approximation, 
 
 These calculations may be performed graphically. For this 
 purpose the functions f(x, y, z) and g(x, y, z) must be given in 
 some handy form. We notice that in our formulas the first 
 argument assumes the values x, x + A/2, x + h. In the next 
 step where x -\- h, y -{- k, z -\- I are the coordinates of the starting 
 point that play the same part that x, y, z played in the first 
 step, we are free to make the change of the first argument the 
 same as in the first step, so that in the formulas of the second 
 step it assumes the values x + h, x + f A, x + 2h and so on for 
 the following steps. All the values of the first argument can 
 thus be assumed equidistant. Let us denote these equidistant 
 values by 
 
 .TO, xi, x 2 , x 3 , 
 
 The values of f(x, y, z) and g(x, y, z) appear in all our formulas 
 only for the constant values 
 
 X = X , Xi, X 2 , 
 
 For each of these constants / and g are functions of two inde- 
 pendent variables and as such may be represented graphically
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 145 
 
 by drawings giving the curves / = const, and g = const., each 
 value of x corresponding to a separate drawing. These drawings 
 we must consider as the graphical form in which the differential 
 equations are given. It may of course sometimes be very tire- 
 some to translate the analytical form of a differential equation 
 into a graphical form, but this trouble ought not to be laid to 
 the account of the graphical method. 
 
 The method now is similar to that used for the differential 
 equation of the first order, y and z are plotted as ordinates in 
 the same system in which x is the abscissa. Equidistant parallels 
 to the axis of ordinates are drawn 
 
 x = 
 
 etc. 
 
 x = x\, x 
 
 On the first x = XQ we mark two points with ordinates yo and ZQ, 
 and from the drawing that gives the values of f(xo, y, z) and 
 
 Fia. 94. 
 
 g(x , y, z) as functions of' y and z we read the values f(x , y , z ) 
 and gfa, y , z ) and draw the lines from x , y , and x , z to the 
 points 
 
 2, yo + h and z 2 , z + h. 
 
 The intersections of these lines with the parallel x = x\ furnishes 
 the points 
 11
 
 146 GKAPHICAL METHODS. 
 
 , 
 
 xi, 2/o + -^ and 
 With these ordinates we find from the second drawing the values 
 
 .( ,h k\ ( ,h ,k\ 
 
 f [ xi, yo + 2", 2 + 2 ) and I *i 2/o + -j , *o + 2 ) > 
 
 and by their help we can draw the lines from XQ, yo and XQ, ZQ 
 to the points 
 
 and 
 
 The intersections of these lines with the line x = xi furnishes the 
 points 
 
 *i, yo + ^ and ari, z + 2 > 
 
 and with these ordinates we find the values 
 
 ./ ,fc . 4\ / , fe 
 
 / I *i, 2/o + ^-, 2 H- g ) , f7 1 a-i, 2/o + -^ , 
 
 which enable us to draw the lines from XQ, yo and XQ, ZQ to Xz, 
 yo + h and a%, Z Q + 1 3 . 
 
 With these two ordinates we find from the third diagram (a; = #2) 
 the values 
 
 /(a&, yo + ^3, 20 rf 4) and gfa, y + Ar 3 , z + 4), 
 
 which finally enable us to draw the lines from x yo and a-oZo to 
 %, yo + h and x 2 , z + ^4. 
 
 On the vertical line x Xz we thus obtain four points, BI, B 2 , 
 B 3 , #4, corresponding to y + ki, y + fa, yo + k 3 , y + A: 4 and 
 four points, BI, B 2 ', B 3 ', B*, corresponding to z + li, ZQ + k, 
 zo + k, 20 + k (Fig. 94). 
 
 5 2 5 3 and 5iB 4 are bisected by the points d and C 2 ; 5 2 '5 3 ' 
 and Bi'Bt by the points d', C 2 '. Finally dC 2 and Ci'C 2 are 
 divided into three equal parts and the points B and E' are found 
 in the dividing points nearest to Ci and C\. 
 
 The same construction is then repeated with B and B' as 
 starting points and furnishes two new points on the vertical
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. 147 
 
 x = 0-4 and so on. To test the accuracy the construction is 
 repeated with intervals of x of double the size. The difference 
 in the values of y and of 2 found for x = x^ enables us to estimate 
 the errors of the first construction they are about one-fifteenth 
 of the observed differences. 
 
 Both methods are without difficulty generalized for the integra- 
 tion of differential equations of any order. We can write a 
 differential equation of the nth order in the form 
 
 or in the form of n simultaneous equations of the first order 
 
 dx 
 
 dt = Xl > 
 
 dx, 
 
 fa 
 
 f(t, X, 
 
 A more general and more symmetrical form is 
 
 dx 
 
 - = fi(t, x, xi, - z n _i), 
 
 The functions x, Xi, a&, a:_i are then represented as ordinates 
 to the abscissa i, so that we have n different curves. When the 
 function f(t, x, xi, Xz, - x^i) is given in a handy form, so that
 
 148 GRAPHICAL METHODS. 
 
 its value may be quickly found for any given values of t, x, x\, 
 - Zn_i, there is no difficulty in constructing n curves whose 
 ordinates represent the functions x, xi, Xz, - x n -i. Starting 
 from given values of t, x, xi, x%, x n -i we have only to apply 
 the same methods that have been explained for the first and the 
 second order.
 
 COLUMBIA UNIVERSITY PRESS 
 
 Columbia University in the City of New York 
 
 The Press was incorporated June 8, 1893, to promote the publication 
 of the results of original research. It is a private corporation, related di- 
 rectly to Columbia University by the provisions that its Trustees shall be 
 officers of the University and that the President of Columbia University 
 shall be President of the Press. 
 
 The publications of the Columbia University Press include works on 
 Biography, History, Economics, Education, Philosophy, Linguistics, and 
 Literature, and the following series : 
 
 Columbia University Contributions to Anthropology. 
 
 Columbia University Biological Series. 
 
 Columbia University Studies in Cancer and Allied Subjects. 
 
 Columbia University Studies in Classical Philology. 
 
 Columbia University Studies in Comparative Literature. 
 
 Columbia University Studies in English. 
 
 Columbia University Geological Series. 
 
 Columbia University Germanic Studies. 
 
 Columbia University Indo-Iranian Series. 
 
 Columbia University Contributions to Oriental History and 
 Philology. 
 
 Columbia University Oriental Studies. 
 
 Columbia University Studies in Romance Philology and Liter, 
 atnre. 
 
 Adams Lectures. Carpentiw Lectures. 
 
 Julius Beer Lectures. Hewitt Lectures. 
 
 Blumenthal Lectures. Jesup Lectures. 
 
 Catalogues will be sent free on application. 
 
 LEMCKE & BUECHNER, Agents 
 
 30-32 WEST a 7 th ST., NEW YORK
 
 COLUMBIA UNIVERSITY PRESS 
 
 Columbia University in the City of New York 
 COLUMBIA UNIVERSITY LECTURES 
 
 ADAMS LECTURES 
 
 Graphical Methods. By CARL RUNGE, Ph.D., Professor of 
 Applied Mathematics in the University of Gottingen ; Kaiser 
 Wilhelm Professor of German History and Institutions for the 
 year 1909-1910. 8vo, cloth, pp. ix-j-148. Price, $1.50 net. 
 
 JULIUS BEER LECTURES 
 
 Social Evolution and Political Theory. By LEONARD T. 
 HOBHOUSE, Professor of Sociology in the University of London. 
 12mo, cloth, pp. ix-f 218. Price, $1.50 ret. 
 
 BLUMENTHAL LECTURES 
 
 Political Problems of American Development. By 
 
 ALBERT SHAW, LL.D., Editor of the Review of Reviews. 12mo, 
 
 cloth, pp. vii+268. Price, $1.50 net. 
 Constitutional Government in the United States. By 
 
 WOODROW WILSON, LL.D., President of Princeton University. 
 
 12mo, cloth, pp. vii+236. Price, $1.50 net. 
 The Principles of Politics from the Viewpoint of the 
 
 American Citizen. By JEREMIAH W. JENKS, LL.D., Pro- 
 fessor of Political Economy and Politics in Cornell University. 
 
 12mo, cloth, pp. xviii+187. Price, $1.50 net. 
 The Cost of Our National Government. By HEKRY 
 
 JONES FORD, Professor of Politics in Princeton University. 
 
 12mo, cloth, pp. xv+147. Price, $1.50 net. 
 The Business of Congress. By HON. SAMUEL W. MCCALL, 
 
 Member of Congress for Massachusetts. 12mo, cloth, pp. vii+ 
 
 215. Price, $1.50 net. 
 
 CARPENTIER LECTURES 
 
 The Nature and Sources of the Law. By JOHN CHIPMAN 
 GRAY, LL.D., Eoyall Professor of Law in Harvard University. 
 12mo, cloth, pp. xii+332. Price, $1.50 net. 
 
 World Organization as Affected by the Nature of the 
 Modern State. By HON. DAVID JAYNE HILL, American 
 Ambassador to Germany. 12mo, cloth, pp. ix+214. Price, 
 $1.50 net. 
 
 The Genius of the Common Law. By the KT. HON. SIR 
 FREDERICK POLLOCK, Bart., D.C.L., LL.D., Bencher of Lincoln's 
 Inn, Barrister-at-Law. 12mo, cloth, pp. vii+141. Price, $1.50 
 net. 
 
 LEMCKE & BUECHNER, Agents 
 
 30-32 West 27th Street, New York
 
 COLUMBIA UNIVERSITY PRESS 
 
 Columbia University in the City of New York 
 COLUMBIA UNIVERSITY LECTURES 
 
 HEWITT LECTURES 
 
 The Problem of Monopoly. By JOHN BATES CLARK, LL.D., 
 Professor of Political Economy, Columbia University. 12mo, 
 cloth, pp. vi+128. Price, $ 1.50 net. 
 
 Power. By CHARLES EDWARD LUCRE, Ph.D., Professor of Mechan- 
 ical Engineering, Columbia University. 12mo, cloth, pp. vii+ 
 316. Illustrated. Price, $2.00 net. 
 
 The Doctrine of Evolution. Its Basis and its Scope. By 
 HENRY EDWARD CRAMPTON, Ph.D., Professor of Zoology, 
 Columbia University. 12mo, cloth, pp. ix+311. Price, $1.50 net. 
 
 Medieval Story and the Beginnings of the Social Ideals 
 of English-Speaking- People. By WILLIAM WITHERLE 
 LAWRENCE, Ph.D , Associate Professor of English, Columbia 
 University. 12mo, cloth, pp. xiv+236. Price, $1.50 net. 
 
 JESUP LECTURES 
 
 Light. By KICHARD C. MACLAURIN, LL.D., Sc.D., President of the 
 Massachusetts Institute of Technology. 12mo, cloth, pp. ix-f 251. 
 Portrait and figures. Price, $1.50 net. 
 
 Scientific Features of Modern Medicine. By FREDERIC 
 S. LEE, Ph.D., Dal ton Professor of Physiology, Columbia Uni- 
 versity. 12mo, cloth, pp. vi-f 183. Price, $1.50 net. 
 
 Lectures on Science, Philosophy and Art. A series of 
 twenty-one lectures descriptive in non-technical language of the 
 achievements in Science, Philosophy and Art. 8vo, cloth. 
 Price, $5.00 net. 
 
 Lectures on Literature. A series of eighteen lectures on liter- 
 ary art and on the great literatures of the world, ancient and 
 modern. 8vo, cloth, pp. viii+404. Price, $2.00 net. 
 
 Greek Literature. A series of ten lectures delivered at Columbia 
 University by scholars from various universities, in the spring of 
 1911. 8vo, cloth, pp. vii+306. Price, $2. 00 net. 
 
 LEMCKE & BUECHNER, Agents 
 
 30-32 West 27th Street, New York
 
 
 1G LIBRARY 
 

 
 >, ^>ffiGO' V E