Columbia Onitoergitp 3lf cturf tf
GEAPHICAL METHODS
ERNEST KEMPTON ADAMS RESEARCH FUND
1909-1910
COLUMBIA
UNIVERSITY PRESS
SALES AGENTS
NEW YORK :
LEMCKE & BUECHNER
30-32 WEST 27TH STREET
LONDON :
HENRY FROWDE
AMES CORNER, E.G.
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HENRY FROWDE
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-
COLUMBIA UNIVERSITY 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
JReto
COLUMBIA UNIVERSITY PRESS
1912
PRESS OP
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
J
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 IT 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
T n -4- 7/ n 7 n
x T y - z ,
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-
v
271620
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.
GRAPHICAL 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
2
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 i.
GRAPHICAL CALCULATION. 6
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.
GRAPHICAL 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 b 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
GKAPHICAL CALCULATION. 5
the line constructed in the way shown in Fig. 3 and Fig. 4 to
the fixed line is equal to the number
ac
b
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 6, 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
ac
x = ~b
whatever the signs of o, 6, 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.
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 ao, a\ 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 #o> Ui) &L> 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
FIG. 5.
a + ai + 02 + + a,
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 Ci, ai from
Ci to C 2 , a n from C n to C n +i-
Let x 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 a , a b a. The length Ox is then defined by the value
GRAPHICAL CALCULATION.
of the ratio x. Through x and 1 draw lines parallel to the ?/-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 ?/-axis in D n , the triangle (7 n Z) n P n _i will be
similar to C n C n +iB n and their ratio is equal to x, therefore
C n D n = a n x. Consequently the height of P n _i above C n -i is
equal to C n -iD n = a n x + a n -i. Now let us repeat the same
operation in letting the point D n take the part of C n+ i. Through
D n draw a line parallel to Ox, that intersects the parallels through
x and 1 in P n _i and B n -\. Draw the line B n -iC n -i that intersects
the parallel through x in P n -2-
Then the height of P n _ 2 above
C n -i will be equal to
Cn-iD n X = (a n x + a n -i)x,
and the height above (7 n _ 2 will be
equal to
:
a n x
a n _ 2 .
B 3
Continue in the same way. Draw
Pn-zBn-z parallel to Ox, draw
B n - 2 C n -2 and find the point P n -s-
Then the height of P n -3 above C n -z will be
(a n x* H
and the height of P n -s above
FIG. 6.
Finally a point PO is found (see Fig. 6 for n = 4) by the inter
section of Bid with the parallel to the 2/-axis through x, whose
height above is equal to
ax
a
Let us designate the line xPo by y, so that
8 GRAPHICAL METHODS.
y = a n x n + a n -ix n ~ l + + a\x + 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 , dn-ix 11 1 , - a\x, a .
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 + ttn-i, a n x 2 + a n -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\ t 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
ao + ai + 02 + + On,
(7 a +i 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 PQ will then represent the
curve, whose equation is
y = a + aix + - + 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 x 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 B a C a 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
A a = a a -10 ft .
Then as
GRAPHICAL CALCULATION.
we find
y = A + AiX + A 2 X 2 + + A n X\
Lay off the lines A , AI, A n in a convenient scale and let
X 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 Ci, 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 f , 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 a are best written in rows in this way
a n a n -i a n -2 a\ CLQ
a n x a n -i x - - az x a\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.
and let all coefficients in the first instance be positive.
The coefficients a , i, 02, a 3 , <z 4 are supposed to be represented
by straight lines, while x will be the ratio of two lines. The lines
do, cti, 0%, ds, tt 4 are laid off in a
broken line do to the right from
Co to Ci, di upward from Ci to
Cz 9 0% to the left from C 2 to Cz, a 3
downward from C 3 to C 4 , a 4 again
to the right from C 4 to C 5 (Fig. 7).
Through (7 5 draw a line C&A to
a point A on C 3 (7 4 or its prolonga
tion and let x be equal to the
FJQ 7 ratio C\A : C^C^ taken positive
when CA has the same direc
tion as C 3 C 4 . Then we have
and
/"Y A I
LsA = a 4 x -f- 3 .
C\A and C$A are positive or negative according to their direction,
being the same as the direction of (7 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 2 C 3 or its prolongation. Then we have
and
C 3 B =
a 3 ) x
GRAPHICAL CALCULATION.
11
ax + CL^X + 02.
C^B and C%B are positive or negative according to their direction
being the same as the direction of C^Cs or opposite to it. Simi
larly we get
and finally
C E =
a .
I
FIG. 8.
is positive, when E is on the right side of CQ and negative
when on the left side. When the point A moves along the line
ftft, the point E will move
along the line ftft 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, /
that we pin onto the drawing
at ft, so that it can freely be
turned round ft. Following
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 a s is
negative ft ft would have to lie above ft; but C^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 ft. 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 C Q C 3 as diameter. Its intersections with CiC z
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
C 2
/ \
y = a Q + ai(x p) + (h(xp)(x q)
+ a 3 (x - p)(x - q)(x - r) +
Let us again suppose CLQ, a\, 0%,
FlQ 9 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 + 1, r and r + 1 on the
axis of abscissas, Fig. 10 is chosen arbitrarily and the position
2 r x p+\ q\
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.
GKAPHICAL CALCULATION. 13
Draw parallels to the t/-axis through p, q, r, x, p + 1, q + 1,
r + 1. On the parallel through r+1 find the point Q of the
same ordinate as C* and on the parallel through r find the point
AQ of the same ordinate as 3. Join AQ and Qo by a straight
line and find its intersection PI or that of its prolongation with
the parallel through x. The height of PI above Cs or A Q is
equal to az(x r) and the height above C 2 is equal to a z (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 C 2 . 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 P 2 above 0% or A\ is equal to
[a s (x r) + (h}(x q),
and the height above C\ is equal to
a z (x r}(x q) + a%(x q) + cti.
Finally find a point Q 2 on the parallel through p + 1 of the
same ordinate as P2 and a point A 2 on the parallel through p of
the same ordinate as C\. Join A 2 and Q 2 by a straight line and
find its intersection P 3 or that of its prolongation with the par
allel through x. The height of P 3 above Ci or AZ will then be
equal to
[ctz(x r)(xq) + (h(x - q) + ai](x - p)
and the ordinate of P 3 will be equal to the given integral function
y = a z (x - r)(x - q)(x - p) + (h(x - q)(x - p)
+ ai(x p) + ao.
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 = g/10, R = r/10 Z = r/10.
We then write
= a , Ai = lOoi, ^ 2 = 10002, A 3 = 1000a 3 ,
14
GRAPHICAL METHODS.
and obtain
y = A Q + A,(X - P) + A 2 (X - P)(X - Q)
+ A S (X - P)(X - Q)(X - R).
The scale for the lines A , AI, A 2 , A s 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, 5;
find the lines o, 0i, (h, &z, 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, g, r, s by y p , y q , y r) y s 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 = + ai(x - p) + (k(x p)(x q) + a*(x p)(x-q)(x-r)
reduces to ao. Therefore we have y p = do. The point C\ is
found by drawing a parallel to the axis of abscissas through P
and taking its intersection with
the axis of ordinates.
In order to find C% draw a
straight line through P and Q
and find its intersection A with
the parallel through p + 1 (Fig.
11). A parallel to the axis of
abscissas through A intersects
the axis of ordinates in C*. For
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
Q
/
A
Ca
^
/
I
,/
j
.2
> ^
1
FIG. 11.
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
p
FIG. 12.
this parallel with the vertical through x let us call X and its
ordinate y f . Then we have
i + 02(3 - g) + osfc - g)(* - r).
a: - p
Let us carry out this construction not only for x = g but also
for a: = r and # = 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 = (oo + a>i) + <h(x q) + a 3 (x 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 f , 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.
y" -
x q
for x = r and x = s, or
y" = a Q + ai + 02 + flsfc - r).
The horizontal through R" intersects the axis of ordinates in the
point 63. Finally we find d 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 CiCzCsC* we can now for any value
of x construct the ordinate
y = a + ai(x p) + (h(x - p)(x - q)
+ a*(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 -f- 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 q -\- 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 f by connecting
R and S with P and drawing horizontals through the inter
sections A r , A 8 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 r ). On the vertical through any point x take the inter
section with R"S" and pass horizontally to the point A x f 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 x 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
y = R(x).
For a rational function can always be reduced to the form of a
quotient of two integral functions
R(x) = gi(x)/gz(x).
Now after having constructed curves whose ordinates give the
values of gi(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 Gz
be the points whose ordi
nates are equal to g\(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 ft. Then R(x) is equal to the ratio Gi A
to PO. Gi A may then be set off as ordinate on the vertical
3
Xlf
a; Axis .
FIG. 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, xz x nt
a + aixi + 04X2 + ---- f- a n x n ,
where a , ai, 0%, a n are given numbers positive or negative.
The question is how the value of this linear function may be
conveniently constructed for various systems %i, x 2 , 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
a Q begins at and runs to Ci, 02 begins at d and runs to C 2 and
so on (Fig. 14). The numbers x\ t x%, x n let us represent
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 01/PO, a* = 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 OAoAiAzA^Ai in such a manner that A is on the vertical
through Ci and OA is parallel to PO, A\ on the vertical through
Cz and A Ai parallel to PI, At on the vertical through Cs and A\A Z
parallel to P2 and so on. Then the ordinate y of A will have
the same length as ao and will be directed upward when the
direction of a is to the right, and downward when the direction
of O,Q is to the left. The difference y\ y Q of the ordinates of A\
and AQ is equal in length to CL\XI, as y\ y Q and a\ have the same
ratio as 01 and PO. A\ will be above or below AQ according to
the line aiXi being directed to the right or to the left and it is
understood that a\x\ has the same direction as a\ for positive
FlG. 15.
values of Xi and a direction opposite to i for negative values
of Xi. Thus the ordinate y\ has the same length as the line
a + a\x\ and its direction is upward or downward according to
the direction of the line a + a\x\ being to the right or to the left.
In the same way it is shown that the ordinate y z of the point A z
is equal in length to
a + aiXi + 02^2,
and 2/3 to
a Q + 0,1X1 + 02X2 +
and so on, the direction upward or downward corresponding
to the positive or negative value of the linear function.
If the values of xi f #2, x n satisfy the equation
+ a 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 C n+ i coincide the equation is satisfied. Consequently if we
know all the values but one of the numbers xi, z 2 , x n the
unknown value can be found graphically. For suppose x$ to be
/IT
An-i
<J\ Ca
Cn-iCn CWi=4n
FIG. 16.
the unknown value we can, beginning from 0, find the broken
line as far as A% and beginning from the other end A n we
can find it as far as AS (Fig. 15). A parallel to A^Az through P
furnishes the point 3 on the axis of ordinates. If xi, x%, x n -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 On/Oo. In
Figs. 15 and 16 all the coefficients a^ ai, , are positive. A
negative coefficient #5 is shown in Fig. 17. The only difference
is that CQ lies to the left of C& and consequently the broken line
passes from AI back to A 5 .
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 OAdA\A<i instead of OAoA
The ordinate y a r of the point A a r is evidently
00
01
+ ...+
Oa
P O
and therefore
PO
P O
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-
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
00
A linear equation with only one unknown quantity
do + CLiXi =
is solved by drawing a parallel to A Q Ai through P. Let a second
equation be given with two unknown quantities
b + bixi + 62*2 = 0.
The lines 6 , &i, 2 are laid off as before. Knowing xi as the
solution of the first equation we can construct the broken line
OB Bi corresponding to the second equation and as #2 must
22 GRAPHICAL METHODS.
coincide with the end of b 2 , we can draw a parallel to ftft
through P and find x 2 . In a similar manner we can find z 3
from a third equation
C + CiXi + C 2 X 2 + C 3 Z 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 Let us begin with two
~l -"-ft "! -A2
equations and two un-
known quantities.
tr \ v - - oo+M+ _ ,
FlG -
= 0.
The lines a , i, <h are laid off on a horizontal line OA AiA 2 and
the lines bo, 61, b 2 on another horizontal line O f B BiB 2 (Fig. 19).
Now let us join and , A and # , AI and ft, ^4 2 and B 2 by
straight lines and let us draw a third horizontal line intersecting
them in the points 0"CQCiC 2 . These points correspond to a
certain linear function
C + CiXi + C 2 X 2 ,
and it can be shown that it vanishes when x\ and x 2 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
i * /L ^ . * r
Co = o + -y (oo o) = y a + 7 &
GRAPHICAL CALCULATION. 23
For a parallel to 00 through A defines with the line AoB Q on
the third and second horizontal line segments equal to CQ ao
and bo o and as these segments have the ratio h/l, it follows
that
, h . k . h
CQ = 0o + y (0o ao) = y o + y 0o-
By drawing a parallel to ^4o#o through A\ and to ^4i#i through
AI or through $ 2 (which comes to the same thing), we convince
ourselves in the same way that
. h , N k h -
Ci = Oi + y (61 - tti) = y i + y 63
and
, h . k h
2 = 02 + y (0 2 02) = y 02 H- y o 2 .
Multiplying the equation
a + ai#i + 02X2 =
by k/l and the equation
60 + b&i + 62^ =
by h/l and adding the two products, we obtain
C + CiXi + C 2 X2 = 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 c , Ci, c 2 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 ^2^2. In this case the points C\
and C^ must coincide and consequently c^, must vanish. If Ci
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 GRAPHICAL METHODS.
lines. In case ci also vanishes, that is to say, in case the three
straight lines A 2 B 2 , AiBi, A B 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 + 2X2 = 0,
and as ci and <% are zero CQ would have to be zero, which it is not
as 00 is supposed not to pass through the intersection of A 2 B 2 ,
AiBi and A B . If on the other hand all four lines A 2 B 2 , AiBi,
A Bo, 00 pass through the same point, Co, Ci and c 2 will all three
vanish. In this case the two given equations do not contradict
one another, but &o&A will be proportional to ao^ic^. The
- j .
A k \ \ n \
/ \ \ \ \
Bo ^i \B S \f*fBi Vfls
C& C 6
FIG. 20.
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
Xi, 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
OAoAiA 2 A 3 A^A 5 A 6 and BoBiB 2 B 3 B^B 5 B 6 represent two linear
equations. Through the intersection of A 3 B 3 and A^B^ a third
horizontal line is drawn intersecting the lines 00 , A Bo, AiBi,
A 6 B& in 0"CoCi C & . As C 3 and 4 coincide, the line C4
vanishes and # 4 is eliminated, so that the equation assumes the
form
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
zi = xi + yii and Zz =
we understand the complex number
+
where
and we write
3= xi + xz and y 3
23 = 2i + 22.
+ y 2 ,
Graphically we obtain the point Z$ representing z z from the
points Zi and Z 2 representing z\ and z 2 by drawing a parallel
to OZ 2 through Z\ and making ZiP
(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 OZ\ in length and direc
tion. The coordinates of P are
evidently equal to x\ + # 2 and
FIG. 21.
Two complex numbers z and z
are called opposite, when their sum
is zero.
z + z or x = x and y = / or z = z .
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
GRAPHICAL CALCULATION.
27
Therefore
22 = (xi - xt) + (i/i - yfii.
This may also be written
zi + z 2 where Z2 /= Z2 = x% y&.
That is to say, the subtraction of the complex number Z2 from zi
may be effected by adding the opposite number 22 . For the
geometrical construction of the point Z corresponding to zi Z2
we have to draw a parallel to OZ 2 through Zi and from Zi in
the direction from Z 2 to we have to lay off the distance Z 2 0.
Or we may also draw from a line equal in direction and in length
to Z 2 Zi. This will also lead to the point Z representing the
difference zi 2fc.
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 f y = r sin <p
and
z = x -\-yi- r(cos<p+ sin<pi).
Let us call r the modulus
and <p the angle of z. The an- FlG 2 2.
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.
28 GRAPHICAL METHODS.
According to Moivre s theorem, we can write
2 = I***.
By the product of two complex numbers
Zi = ne* 1 * and Zz
we understand that complex number 2 3 whose modulus r 3 is
equal to the product of the moduli r\ and r 2 and whose angle ^
is the sum of the angles <pi and <pz or differs from the sum only by
a multiple of four right angles
23 = z& =
The definition of division follows from that of multiplica
tion. The quotient 21 divided by Zz is that complex number,
which multiplied by Zz gives z\. Therefore the product of its
modulus with the modulus of Zz must be equal to the modulus of
zi and the sum of its angle with the angle of Zz must be equal to
the angle of 21. Or we may also say the modulus of the quotient
2i/Z2 is equal to the quotient of the moduli ri/r 2 and its angle is
equal to the difference of the angles <pi <pz. 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
= 2 3 /2 4 .
The geometrical meaning of this is that
= r 3 /r 4 ,
and
<Pl <P2 = <P3 <?*
That is to say, the triangles ZiOZ 2 and Z 3 OZ 4 are geometrically
GRAPHICAL CALCULATION.
29
similar (Fig. 23). When three of the points Zi, Z 2 , Z 3 , Z 4 are
given the fourth can evidently be found. For instance let
Zi, Z 2 , 4 be given. Draw a parallel to ZiZ 2 intersecting 0Z 2
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^Z Z 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 0Z 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/2fc = 2 3 Or Zi = SfcZs-
From any two of the points Zi, 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 = 2Tr/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) + f sm(nt + <*)*
or
z = re (nt+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 2 , we can write
z = z e nti .
The geometrical meaning of the product
z e nti
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 ZQ is not
changed by the multiplication. The
movement of the point Z therefore
t=r,
A
consists in a uniform revolution of
f=0 OZ round 0. At the moment t=
\ the position is OZ Q and after the
* 1 *- time T 2w/n the same position is
\ / occupied again. The revolution goes
-v \ / on in the direction from the positive
x -- I - *=% axis of x to the positive axis of y
(Fig. 25).
FIQ 25 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 = n cos (nt + i) + **2 cos (nt + 2 ),
and let us again substitute the motion of the point Z correspond
ing to the complex number
For t = the first term is
and the second term
Introducing Zi and 22 into the expression for z we have
2 = Zie + 226?*" = (zi + &)e nti = z 3 e n <*
where
23 = Zi + 22-
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 0Z 3 corresponding to the
complex number
23 = zi + 22.
The projection of Z on the axis of x has the abscissa
x = TS cos (nt + as)
where r 3 and 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 n, r 2 , r x ; ai, a 2 , a x and w are constants may be
written
or
32 GRAPHICAL METHODS.
2 = zoe nti ,
where
20 = 2i + 22 + + 2 A .
The movement of Z therefore, excepting the case ZQ = 0, consists
in a uniform revolution of OZ round 0, OZ always keeping the
same length equal to the modulus of 2 . The position of OZ at
the moment t = is OZ .
The motion of a point P whose abscissa is
x = ae~ kt cos (nt + 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
2 = ae~ kt cos (nt + a) + ae~ kt sin (nt + d)i,
or
or
2 = z e- kt -e nti ,
where 2 is written for the complex constant ae ai .
The product
is a complex number corresponding to a point Zi on the same
radius as Z Q , coincident with ZQ at the moment t = but ap
proaching in a geometrical ratio after t = 0. In unit of time
the distance of Z : from decreases in the constant ratio e~ k : 1.
The multiplication with e nti turns OZ\ 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.
33
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 r units of time later, we
shall have for the first interval
z = z Q 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 z\ and z%
the corresponding values of z
at the moments t\ + T and U + T of the second interval, we have
FIG. 26.
Therefore the triangle Z\OZ<i is geometrically similar to the
triangle ZiOZ 2 . As Z\ and Z 2 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
or
= (- k + ni)e- kt e nti = (- & + ni)z
dz
dt
= k + m =
z
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 f is directed vertically upward or downward or when the
angle of dz/dt is equal to 7r/2 or 3ir/2. The angle of z will then
be 7T/2 X or 3?r/2 X plus or minus any multiple of 2r. As
the angle of z, on the other hand, is changing in time according
to the formula
nt+ a,
we find the moments where the movement turns by the equation
nt + a = 7T/2 - X + 2N-7T,
or
nt + a = 37T/2 - X + 2Nir t
N denoting any positive or negative integral number. The time
between two consecutive turnings is therefore equal to v/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 3r/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~--
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.
GKAPHICAL 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
^ I ], + n 2~ = F e Pti
dt 2 ^ dt^
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
*o(- mp 2 + kpi + n 2 ) = F,
or
F
mp 2 + kpi + n 2
ZQ is a complex constant, that may be represented geometrically
as we shall see later on.
This solution
z = z Q e pti
is not general. If z r denotes any other solution so that
36 GRAPHICAL METHODS.
we find by subtracting the two equations
or writing
z z = u,
d?u du
The general solution of this equation is
u = u^ + u^e^y
where ui and u^ are arbitrary constants and Xi and \2 are the
roots of the equation for X
mX 2 + k\ + n 2 = 0,
2m"
If & 2 /4ra 2 is greater than n 2 , so that the square root has a real
value, l/& 2 /4m 2 n 2 will certainly be smaller than k/2m. There
fore Xi and \2 will both be negative and the moduli of the complex
numbers u\e K * and utf K * will in time become insignificant. If,
on the other hand, & 2 /4m 2 is smaller than n 2 , both complex
numbers u\e Klt and u^* 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 = z Q e 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 Q , completing one revolution in the
period Zir/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 = TQ COS (pt + a),
where r is the modulus and a the angle of z . 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
z - -
let us consider the denominator
mp 2 + kpi + n 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 +
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
p} y _
viz.,
38
GRAPHICAL METHODS.
n
2_,
k 2
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
>=0, 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
and consequently
FIG. 27.
and
= re*
- F -t
r
x
F
- cos (pt
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 away from 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
but there will be no definite period for which the forced vibra
tions are stronger than for all others.
CHAPTER II.
THE GKAPHICAL REPRESENTATION OF FUNCTIONS OF ONE OR
MORE INDEPENDENT VARIABLES.
6. Functions of One Independent Variable. A function y of
one variable x
y =
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 = (f>(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-
FIQ. 30.
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,
"a?
, 12. These ordinates as a rule will not be equidistant. But
as soon as we move them so as to make them equidistant, they
form the ordinates of the curve
1-
-6
I I
=/(*>))
with t as abscissa (Fig. 31).
The representation of a func
tion by a scale may be general-
k 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
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 x-scale through an amount y = c so that a
point of the x-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
f(x) = g(t) + c.
If x, t and x , t , denote two pairs of values that are placed
opposite to one another, we shall have simultaneously
/Or) = g(t) + c,
/(* )
or by eliminating c
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 x
t " t
Li 1 V f V , 3
.* , 5 i 6 , 7 | 8 , 9 f 1
> t
i 5 i
2 2 ! 5 ! 3 Qj
i V P 8 ! 9 Jo I
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 y
I i M ? ? i^prmo v y [ 3p ^jff
L5 2 4156 78910 15 20 30 40 60 801"
*-. 5 ? ? f f ? T ? ?y
1.5 2 3 1 5 678 9 1*0 i
FIG. 34.
then t is read off opposite to x f . 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 7 .
By means of a little frame carrying a crossline and sliding over
the instrument, we can bring the scales x and T 7 or t and X op
posite each other. If, for example, for any position of the
instrument x, I 7 and x , T are two pairs of values opposite each
other, then
log x - 2 log f = log x - 2 log T ,
or
If any three of the four quantities x, T 7 , 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 7 . Or we can find the value of
by setting z opposite to T and reading off the value opposite x .
GKAPHICAL 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 i and XT = ZT,
so that any two values opposite to one another have the same
product.
For x and T we have
log x + 2 log T 7 = log x + 2 log T 7 ,
or
Let us apply this to find the root of an equation of the form
u? + au = 6.
Divide by u so that
U
and set T 7 = 1 opposite to X = 6. Then taking T = u we find
on the same cross line t = u 2 and Z = b/u, so that we read the
two values u 2 and b/u 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* - 5u = 3,
or
u 2 -5=-.
u
We set T = 1 opposite X 3 and run our eye along the scales
X and t (Fig. 36), to find the place where t 5 = X. We find
X H 1 ,2
3 4 5 6 ?? 910 20 ,
r *
) Of 08 OjS OT6SI 9 G 1
, S 1 T " i ^ T
LLJ ]
JC >\ i 4 i
5 2 2^5 3 4 5
7 8 9 10 1
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/u is negative,
and therefore the positive value of 3/u plus u 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 T-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 2 b/u
GRAPHICAL REPRESENTATION OF FUNCTIONS. 47
runs from oo to + GO 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? + au = 6,
because from a positive value of a it follows that u* + au will
for u = oo to +00, run from oo to + oo 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
iP - = a.
u
For negative values of b we have to investigate whether there
are positive roots. For positive values of u the function u 2 b/u
has a minimum, when the differential coefficient vanishes, i. e. y for
or
U
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 2 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 b 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 w 2 b/u 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
2_ K _ ?
u
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/u 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/u 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(0,
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(t) 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
/
GRAPHICAL REPRESENTATION 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
log * + log log (l + ~j = log f + log log (l + ^~) .
We have only to make the or-scale
y = + log log(l +155)- log log (l + T |) ,
and the /-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 y-scale would have to be 12
cm. In the normal position of the instrument the readings x f 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
7.24.
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 a>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
y= x> y= n-~f
In the normal position of the instrument the scale division
marked x = oo 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 = l/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
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
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 I
1
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
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 / ,
or
_z tf_
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 = <p(u) and y = $(v).
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 = \f/(v) 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 v(u) and \//(v)
FIG. 38.
1 .2 3 4 5
FIG. 39.
to simplify the form of the curve. It is easily seen, for instance,
that a curve representing an equation f(u, 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, v) = now represented by
the straight line.
Suppose we want to represent the relation
where a and b 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 y = $
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 0-scale (Fig. 40).
Any point on the straight
line corresponds to four
combinations -\-u, +0; u,
ti+i.5j* 2.5 3 5 it"5o*) + ; u, v; u, v, be
cause x has the same values
for opposite values of u
and y for opposite values of v. We can read v as a function of
w or u as a function of v.
If a second equation
FIG. 40.
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
u*
and
22 r 32
^ 2V
42 + 52
1,
approximately equal to u = =*= 1.2 and = 2.4.
Another function much used in mathematical physics
v = ae
GKAPHICAL KEPRESENTATION OF FUNCTIONS.
55
may also be represented by a straight line by means of the same
device.
By making
y = log v, x = w 2 ,
we obtain
2/ = loga- Jj,
where log v and log a are the natural logarithms of v and a.
The w-scale is laid off on the axis of x and the 0-scale on the axis
of y and we have to join the points u 0, v = a and u = m,
i) = a /e. The point v = o/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.
T-^CK)
FIQ. 41.
Suppose we had to find the constants a and m from two equa
tions
Vi == CL6~m i
and
Our diagram would furnish two points corresponding to u\, v\
and uz, ^. The straight line joining these two points intersects
the axis of ordinates at v = a and intersects the parallel through
v = 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.
10
2.5 3 3.5
FIG. 42.
10
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 = c 1/r on the w-scale with
the point v = c 1/8 on the 0-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
f(u, v) = 0,
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 v.
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 = log v + log b = 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
u, =
58 GRAPHICAL METHODS.
drawn on logarithmic paper is therefore identical with all the
curves
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
* = f(x, 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 z. 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-
GKAPHICAL 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
2* + pz + g =
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 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 read the roots of any equation of the third degree
of the form
s 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.
-oj
1.3
1.2
1.1
10
0.9
0.8
0.7
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/m instead
of z and to choose m sufficiently large.
Instead of
z 3 + pz + q =
GRAPHICAL REPRESENTATION OF FUNCTIONS. 61
we obtain
wY 3 + pmz + q = 0,
or dividing by ra 3 ,
or
2 3 + p z +q = 0,
where
^ = ^ ^ = JL
p m 2 3 m 3
By choosing a sufficiently large value of ra, p f and # can be
made to lie within the limits 1 to + 1 so that the roots z r
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, v, 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 = \j/(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)<f>(u) + b(iv)\f/(v) + c(w) 0,
a, b, c being any functions of w, <p any function of u and \f/ any
function of v.
For introducing
62 GRAPHICAL METHODS.
X = <f>(u), y = ^(fl)
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 8, 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 <p cos 6 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 5 + x cos <f> cos 5.
When x and y are plotted as rectangular coordinates, we obtain
GRAPHICAL 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 = II
FIG. 46.
straight lines 5 = 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 + 5), 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 + 5) and a scale on the vertical x + 1, showing the points
y = cos (<p 6) 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 5 are the complements of h and the values of <p + 5 the
complements of h. For a latitude of 41, for instance, we
have
For 5 <p + 5 <f> 8
June 21 23.5 64.5 17.5
September 23 and March 21 41 41
December 21 . . . . . . -23.5 17.5 64.5
64
GRAPHICAL METHODS.
The values of <p + 5 and <p 8 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 5 changes its position continually as 5 changes
continually. But the changes of 6 during one day are scarcely
appreciable unless the drawing is on a larger scale.
If in the equation
ax + by + 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
5 x=<t>w 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,
0-scale and w-scale (Fig. 47). If we move the transparent material
without turning it and make the first two lines intersect the u-Sind-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.
Fia. 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
^-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, v),
y = }(u, v).
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, v)
and \f/(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-
6
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
<p(u, v) = a and \j/(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
N
3T0.7
p
v*
0.2 0.3 0.4 0.5 0.6 OJ
FIG. 48.
infinitely small distance, u changing to u + du, v to v + dv and
x to x + dx, where
Let us write
dp d<p
dx du-\- dv.
du dv
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-
GKAPHICAL 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(f>/dv and let us write
d<p d<p .
-=rcosX, ^-rsinX,
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 + <fo = refe cos (a X),
cm ofl
or
dx .
-r = r cos (a X).
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
dx .
-7 = r cos (a X)
we see that the ascent is steepest for a = X, where dxlds = 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
dx
5-
and therefore
a - X = 90.
If PN is the gradient of the function \j/(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: ty/du, d^/dv) 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 FIG. 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 + dv; D : u + 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
GKAPHICAL METHODS.
d<p
~
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\p j
dx2 = -j-dv, dy2 = -r afl.
dv dv
Denoting the lengths of AB and AC by dsi and ds 2 and the angles
that the directions of AB and AC form with the direction of the
du
I
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
dx 2 = ds-2. cos 72, dy z = ds 2 sin 72,
or
dsi
and
We may call
T- = cos 71-T-j ^ = sm 71 -j-
du du du du
d<p
du
the scale of depiction at A in the direction AB and
GRAPHICAL REPRESENTATION OF FUNCTIONS. 71
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 dujds\
and dv/dsz.
The area of the parallelogram ABCD in the xy plane is
uft sm * - 71 = -- - - -
According to the way in which the angles 72 and 71 are defined
sin (72 71) 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
d(p d\f/
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.
d<pd\j/
du dv dv du
is called the functional determinant of the functions <p(u, v) and
t(u, v).
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 v, we pass to a point D close by whose coordinates are
u + Aw, v + A0. In the xy plane we find the corresponding
points A and D with coordinates (Fig. 51).
72
GRAPHICAL METHODS.
. X = <f>(u, 0)
y = 4>(u 9 v)
AX =
Ay =
AW, fl
Aw,
We expand according to Taylor s theorem, and writing for
shortness
d<p dtp d\J/ &\f/
<Pu = > <Pv == " 9 TU == > Yv ==
du dv du dv
we find
Ax = <p u Au + <p v Av + terms of higher order,
Ay = \l/ u Au + ^ v Au + terms of higher order.
FIG. 51.
The length of AD and the angle of its direction we denote by
Ar and a in the wo plane and by As and X in the xy plane.
The limit of the ratio As/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
Au = Ar cos a,
Av = Ar sin a,
we obtain
Ax = ((f> u cos a + (p v sin a)Ar + terms of higher order,
ty (&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
~T- == <Pu cos a -j- <p v sin a,
GRAPHICAL REPRESENTATION OF FUNCTIONS. 73
dy
~r^u cos a + \f/ v sin a.
For dz/dr and dy/dr we may also write ds/dr cos X, efo/dr sin X.
ds
~r cos X = <p u cos a -f- <pv sin a,
-7- sin X = \f/ u cos a + \J/ 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 w
z = -T+^r l = T e >
dr dr dr
Zl = <Pu + tut,
22 = <P* + <M-
Multiplying the second of the two equations by i and adding
both they may be written as one equation in the complex form:
z = Zi cos a + 22 sin a.
The modulus of z is the scale of depiction of the uv 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 = %%.
Let us substitute
COS a = - - , Sin QJ = -
J
and write
Zl + ZfcA , Zi Z2/1
so that the expression for z becomes
74 GRAPHICAL METHODS.
z = ae ai + be"**.
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=0- 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
PJQ 52 ae ai and 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 ^ = T\B . o == i
ao corresponds to the direction bisecting the angle between a
and b and ai denotes half the angle between a and b (positive or
negative according to the position of a and b).
or
= (ri + r 2 ) cos (a i) + (ri r 2 ) sin (a ai)i.
Denoting the coordinates of the complex number ze~ a<>i by and rj
we have
= cos (a on) and - = sin (a i),
TI+ r 2
and consequently the equation of an ellipse
(ri + r 2 ) 2 (TI r 2 ) 2
GKAPHICAL KEPRESENTATION OF FUNCTIONS.
75
This ellipse turned round the origin through an angle equal to
<XQ gives us the points corresponding to z. The principal axes
are 2(r + r 2 ) and 2(ri r 2 ) (Fig. 53). The construction of
FIG. 53.
Fig. 53 is obvious. After plotting zi and 22 we find z^/i and
Z2/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 6.
The figure shows that in case a and 6 have the same modulus,
the triangle Z 2 /i, Z\, Z 2 /i becomes equilateral and AZi is per
pendicular to the line joining Z z /i and Z z /i. In this case AZ\
and AZi would have the same or the opposite direction. But as
21 = <f>u + $ui, 22 = <f>v + fai, this would mean that <p u \f/ v <p v \fs 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
GKAPHICAL METHODS.
Z moves on the ellipse from Z\ to Z 2 through the shorter way.
Zi corresponds to a = 180 and Z 2 to a = 270. Now we
have shown above that a positive value of the functional deter
minant <p u \I/v vv^u means that Z 2 is on the positive side of Z\ t
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-
V
D=o
u
FIG. 54.
D
AM
-V*
-1/4
-1/2
-1/1
tions coincide in the way shown in Fig. 54. On this curve the
functional determinant D = <p u \f/ v <p v \ls 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 = #4, x 3) x 2
until at the point A on the curve x = Xi we reach the curve D = 0.
In the xy plane the corresponding path is a parallel to the axis
of # at a distance y\ passing through #4, ar 3 , x 2 and reaching a
point A at x\. If we now proceed on the curve y yi in the
uv plane beyond the curve D = 0, we again intersect the curves
X2, #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 Xz, yi; x 3 , y lt 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
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.
dyfdr
dx/dr
-rl-
FIG. 56.
ds dx
cos X^; = ^- = ^> u cos a + <p v sin a,
ds dy
sin A = = \l/ u cos a + ^ sm a.
u
At the point A we have
Assuming that the gradients at ^4 do not vanish, so that we
can write
<p u = r cos 7 , <p v = r sin 7,
^ M = r cos 7 , &, = r r sin 7 ,
where r and r are positive quantities, the equation <p u ^v~~<pv^u = Q
reduces to sin (7 7 ) = 0, that is, 7 = 7 or 7 = 7 + 180.
It follows therefore that:
ds
cos XT~ = r cos (a 7),
sin X-T- = r cos (a 7 ) = == r cos (a 7).
ctr
Consequently for all directions a in the uv plane for which
cos (a: 7) is not zero, we have
tgX
GEAPHICAL 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
cos X -j- = 0,
dr
ds
sin X T~ = 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 = 7 - 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 uv
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:
It is a maximum in the direction a = 7 or 7 -{- 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.
AB
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 5 with the
ground plan, the plan of ele-
E ^ \ \ G Q vation being parallel to its
\ ^br^T" ax is> 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
x = u
and
1
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, v) in this case are
<p(u, v) = u,
and
i,v) = utg5+- s l/a*-
cos 5
= 0; tu = tg 5,
cos 5 I/a 2
cos 5 I/a 2
GRAPHICAL REPRESENTATION OF FUNCTIONS.
81
c
A\
B
D
u
FIG. 58.
The functional determinant vanishes f or 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 \l/(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
FIG. 59.
82
GKAPHICAL 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
ds
-j-dr,
dr
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. 60.
= <Pu
~ dr
dy ds
dr 1 ~ dr 6 >
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 = Zi cos a + 22 sin a,
GRAPHICAL REPRESENTATION OF FUNCTIONS. 83
or
z = ae ia + be~ ia ,
where
In the case where the scale of depiction dsfdr, 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,
2 = ae ai = -j- e xt .
dr
The complex number a may be written | a \ e a< *, where | a \
denotes the modulus of a and a Q the angle. Both may vary
from point to point, but at every point they have fixed values.
Consequently we have
ds . ,
-j- = I a I and X = a + o.
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 CXQ.
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 = be~ ai . 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.
84
GRAPHICAL METHODS.
and therefore either
Hence in the first case
or
= %2/i.
a =
+
and in the second case
= zi, b = |(zi
a = 0, 6 =
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
y,
\.
-
-
^
.
v
x
^
x
N
\
N
\
\
-*
^
^s,
\,
\
\
\\
x
\]
\1
^/ t
\
\ .
^
>
v<
=000*.
FIG. 61.
GKAPHICAL 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
FIG. 62.
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, 7). 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
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
(i) y
(II). x
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 + fj.
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 = ju
(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 _ mx
A0~~~x
m.
FIG. 64.
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 = I.
If P and Q are the intersections of
these lines with the line on which y
is measured, we have BP = n and PQ = m. Any point in the
plane thus leads to an equation
y = mx + IJL,
except the points on the line on which x is measured. For
m = the equation reduces to
y = M,
that is, the equation of a point on the line on which y is measured.
Instead of y = mx + /*, 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 b.
88 GRAPHICAL METHODS.
Dividing by c another convenient form is obtained,
ax by
T ~
c c
or writing
c _ c
a b
x v
XQ determining the point of intersection of the line BO (Fig. 64)
and the z-line, while y determines the point of intersection of
the line AO with the y-liue.
A curve may be given by an equation
ai(u)x + bi(ii)y + ci(u) = 0,
in which a\(u), b\(u), Ci(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(0)3 + bz(v)y + 02 W = 0,
as(w)x + b 3 (w)y + c 3 (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
oi(6 2 c 3 6302) + 02(6301 ~ 6ic 3 ) + 03(6102 6 2 ci) = 0,
and, vice versa, if the equation between u, v, w may be brought
GRAPHICAL REPRESENTATION OF FUNCTIONS. 89
into this form where a i} bi, Ci are any functions of u, 0%, 62, c z any
functions of v and a 3 , 63, c 3 any functions of w, we can form the
equations
aix + biy + ci = 0,
chx + b 2 y + & = 0,
a 3 x + b 3 y + 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 + c =
remains of the same form, when the units of length are changed
for x and y. If x f 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 = Xz , y = py ,
and the equation of the point may be written
oX* + W + c = 0,
or
a x + by + c = 0,
where a = Xa and & = /z6.
It is sometimes convenient to define the line coordinates in
another way. Let and 77 denote rectangular coordinates
measured in the same unit, then the equation of a straight line
can be written
f\ = tg vt + i?o,
where y 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 77.
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 , rj is expressed by the equation
rj = x + y,
or
y = - & + 77.
If we fix the values of x and y, all the values , 77 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 , 77, and therefore we call it the equation of the
point.
The more general form
ax + by + c =
can be reduced to
a c
y=.-- x -~.
It therefore represents the equation of the point, whose rec
tangular coordinates are = a/6 and 77 = c/b. The case
where b = or
ax + c =
represents the equation of a point infinitely far away in the
direction <p or the opposite direction <p + 180, <p being defined by
c
tg <p = x = - - .
ii
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 = w
may be written in the form
10- v
or
log u + log v = log w,
x + y = log w,
FIG. 65.
when
x = log u and y = log v.
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
x = 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
uv
w
or
log u + log i) = log w
expresses the condition that the three equations
x = log u, y = log v, 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 of w, v, w lie on the same straight line x, y.
The more general relation
yO-yP _ w ^
where a. and 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
straight lines x, y, whose coordinates sat
isfy the equation, for instance
10
FIG. 66.
and
x = log w, y =
0, y=J c log
GRAPHICAL REPRESENTATION OF FUNCTIONS.
93
f-o.5
+1.0
4-0.5
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
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 p , v
and want to find other values
p, v, for which
-0.5
-4.0
-1.5
-2.0
-0.5
-1.0
-L5
-2.0
pv k = p v k .
FIG. 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 po and VQ, so that we need not know the value of
poVQ 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
w 2 + xw + y = 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
# 0, y = w 2 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 x + aW H~ 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 = w 2 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
, w) + h(t, w) =
may be represented in this way, <p(u} denoting any function of
u, \{/(v) any function of v and/0,
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 = <p(u), y = \l/(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 <p the latitude. The angles
7T/2 (p, ir/2 h, 7T/2 8 are the three sides of a spherical
96
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
z the angle PZS.
The equation is
sin 5 = sin <p sin h cos <p cos h cos a.
We write
x = cos a, ?/ = sin 5,
so that the equation becomes
y = sin <p 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 , 77 denote the
rectangular coordinates of the
point, the equation of the points
takes the form
rj = x + y or y = 77 - r,
so that in our case we have
= cos <p cos h, t\ sin <p sin h.
The curves <p = 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 FlG 70
variable value of h we find
COS 2 <f>
+
y
sin 2 <p
GRAPHICAL REPRESENTATION OF FUNCTIONS. 97
and for a definite value of h and a variable value of <p
*> 7 * *) 7 * *
cos 2 h sin 2 A
Any of the ellipses intersects all the others and in this way they
form a network. A point of intersection of the ellipse <p = GI
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 = c 2 identical with h = c 2 (Fig. 70). The easiest way to
find this network consists in drawing the straight lines
+ 77 = cos O h),
and perpendicular to them the straight lines
77 = cos (<p + h),
for equidistant values of <p + 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 d = 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 <p, 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, rj = and mark a scale for the azimuth on it, making
rj = 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
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 8. 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 v, 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 //sin a,
sin h = sin <p sin 5 + cos <p cos 5 cos t.
Substituting these values we find
sin 8 = sin 2 <p sin 6+ sin <p cos <p cos 8 cos tf cos <f> cos 8 sin t ctg a, or
cos 2 <p sin 5 = sin <p cos <p cos 8 cos t cos <p cos 8 sin t ctg a.
Dividing by cos 2 p cos 8 we finally obtain
sin t
tg 5 = tg (p cos t ctg a.
In order to represent this relation graphically we introduce line
coordinates
x = ctg a and y = tg 8
and find
sin t
y = tg <p cos t x.
COS tp
Let us use the second system of line coordinates. The rec
tangular coordinates , 77 of the point represented by the equation
are found from it equal to:
GRAPHICAL REPRESENTATION OF FUNCTIONS.
sin t
99
The curves
^ COS<p
const, are ellipses,
tg <p cos t.
The curves t = const, are hyperbolas,
?
sin 2 1 cos 2
= 1.
The ellipses and hyperbolas are confocal, the foci coinciding
with the points = =*= 1, rj 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 5, the ordinate in both cases
being the tangent of the angle with the
only difference that 6 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
10
FIG. 71.
100 GRAPHICAL METHODS.
hyperbolas the value of t changes abruptly. The line t = 6 h is
meant to start from the focus (j = 1, 17 = 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.
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 f unction /(x),
= ff(x)dx.
Jo.
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 /(or) is a constant the equation
f( x ) = c is represented by a line parallel to the axis of x and
Y = f cdx = c(x - a).
FIG. 72.
Y 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.
y\ 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
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 ci is added, so that the equation becomes
Y = c (x a) + Ci
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 2 to 3,
etc., while it changes its value abruptly at #2, #3> etc. The
line presenting the integral
= ff(x)dx
does not change its ordinate abruptly. It consists of a con
tinuous broken line, whose corners have the abscissas Xz, x z , etc.
DIFFEKENTIAL 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, 0, 7, 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 = x\ (in Fig. 73 x\ is equal to 0) as far as the vertical
x = #2. Through the point of intersection with the vertical
x = x<2 we draw a parallel to P/3 as far as the vertical x = x$.
Through the point of intersection with the vertical x = x% 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
y =
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 /(a*).
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 A\
of the given curve. The vertical part A^Bi and the following
horizontal part BiBz 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 Kb 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 Bi have to be equal. When Kb is fixed the
right position of A%Bi may be found by eye estimate. The eye
is rather sensitive for differences of small areas. Besides a shift
of AiB\ 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^CiCz is
drawn with its vertical part B%Ci 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 A\. 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
T7
and that their points of contact with the integral curve lie on
the same verticals as the points A\ t Kb, 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 Kb, K c , equal to the
integral of the stepping line. Now in the points A if 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 A\A^ 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.
= ff(x)dx
tJa
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
H I
FIG. 74.
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 or, 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
H I
from D. That this is the right position of EH is shown as soon
as we can prove that the area ADBGA is equal to the rectangle
EHBG. The area ADGBA can be divided in two parts, the tri
angle AEG and the part ADBCA between the curve and the
chord. The triangle is equal to the rectangle FIBG, while ADBCA
is equal to two thirds of the parallelogram MNBA, and hence
equal to the rectangle EHIF. 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
V /(X) (Fig. 77), the axis of x and the ordinates corresponding
to x = and x . The statical moment with respect to the
vertical through x = t- is the integral of the products of each
element ydx and its distance x from the vertical
FIG. 77.
M= f (f - x)ydx.
Jo
Let us regard M as a function of and differentiate it:
= +
f ydx.
Jo
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 f or = the
second integration must also begin at the abscissa x = 0.
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 0*1 is the integral
x)ydx.
Considered as a function of Xi its differential coefficient is
* d &
X* d C
-T- Oi - x)ydx = ydx.
axi JQ
That is to say, the differential coefficient is independent of x\,
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
I ydx,
Jo
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
XI
(xi x)ydx
.
for 0*1 = is equal to the statical moment
f 1
Jo
We have therefore only to draw a parallel to PQ through the
last point R of the curve for M(Q found by the second integration.
The ordinates of this straight line for any abscissa xi represent
the values of
,-*
(xi x}ydx
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
I
Jo
ydx
about the axis x = is found in a similar way. It is expressed
by the integral
Jo
Considered as a function of we find by differentiation
dT 2 C* d
= + 2 jf ({ - 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 M(). For = we have T 0, so that the curve
begins on the axis of x at = 0,
The integral
(* x
ydx
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
jfV
we now integrate
ydx + c.
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 = ci 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
XX
fff(x)dxdx
a +C(X-CL)
C(x-a)
ffdx+C
fff(x)dxdx
C
cc
Jf(x)dx
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
I ydx
J a
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 b 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
/
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
FIG. 80.
I
%Ja
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
9
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.
Ja
That is to say, F a is equal to the mean ordinate of the curve
-jf
in the interval x = a to b. 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 b .
Shifting the position of P to P on the horizontal line FG
and repeating the integration
/
t/a
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 to 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 = /2(#)- The whole area is then equal to the difference
FIG. 82.
f ft(x)dx - f f 2 (x)dx,
\J a / a
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
GEAPHICAL 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
FIG. 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.
Q X\ #2 &8 &
FIG. 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.
DIFFEKENTIAL 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 , Kb, - , 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
FIG. 86.
FIG. 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 XQ, ?/o 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 XQ, yo
and at the points where it intersects the curves f(x, 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 X Q
furnishes the differential coefficient df/dx for x = x , 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 GKAPHICAL METHODS.
T r cos a + r sin a. + terms of higher order in r,
&
a is the direction from the point x, y to the new point x + r cos a,
y -\- r sin a and r is the distance of the two points. Dividing
As by r and letting r approach to zero we find
,. A* df cl/ .
lira = cos a + T- sin a.
r d dy
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
Af f)f
= I cos X, = I sin X.
dx dy
Then we have
n_f nf
T~ cos a + sin a = I cos (a X).
O# O2/
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
/ measured in this unfit 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
DIFFEKENTIAL 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
First approximation
2nd approximation
(lutegrationofthe
curve fcelow)
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(l-, rf} 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 (6) 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 2/1 the function of x that corresponds to the curve forming the
first approximation. The second approximation y 2 is then ob
tained as an integral curve of f(x, y\) t that is, of dyi/dx
DIFFERENTIAL AND INTEGRAL CALCULUS. 123
I
Jx n
f(x,yi)dx,
denoting by x p) y p , the coordinates of P. For this purpose the
curve whose ordi nates are equal to f(x, y\) or dyi/dx has to be con
structed first. The values of f(x, y\) 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, y\)
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
Y = f(x, 2/1)
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
Y = f(x, yi)
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
r-/(*,i&)
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
y p + I f(x,
J *
or that
that is to say, that y 2 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
y n +i =
J f(x, y n
The true solution with the same initial conditions y = y p for
x = Xp satisfies the equation
r
Hence
y n+ i - y = J [f(x, y n ) - f(x, y)]dx,
or
Let us now suppose that the absolute value of
f(x, yn) - 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
Me n | x x n |
( | 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-Xn\ k<l,
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 -\, and
so on, so that
e n +i ^ ke n ^ We n -i ^ k n d.
But as e\ 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 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
f(x, y n ) - f(x, y)
y n - y
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
y n+l
f
- y = I
jz
y
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 i i 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
/(a, yn) - f(x, y)
y n - 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 or, 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
/fo yn) ~ f(x, y)
y n - y
for all values of or, ?/, y n concerned. In order to find the maximum
value we may as well consider
SL
dy
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
f(x, y n ) - f(x, y)
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
/(a>yn) /(a, y)
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 ,
i) 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 = ^ = <p(S, 77)
(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 + i) sin co,
y = sin co + 77 cos co,
which are equivalent to
= x cos co y sin co,
77 = x sin co + y cos co,
w 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
jj- = tg (a - co) = f(x, y).
Consequently we obtain for a given value of co
df = _ 1_ _da
dy ~ cos 2 (a co) dy
or remembering that a is given as a function of and 77,
df 1 f da . ] da \
= 5-7 - ; 1 sm co + - cos co ).
dy cos 2 (a co) \ d dy )
For simplicity s sake we shall assume that the axis of is the
tangent of the curve ^(,77) = const, that passes through the
given point, so that da/d = 0.
We then have
df 1 da
and our object is to find how df/dy varies for different values of
DIFFERENTIAL AND INTEGRAL CALCULUS.
129
co. The value of da/drj is independent of w; 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 <p(, rf) = const.
It is no restriction to assume the value of da/drj positive; it
only means that the direction of the positive axis of rj 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 df/dy as an abscissa.
For instance for co = a, df/dy assumes the value
da
cos a.
The abscissa corresponding to this value is AB (Fig. 89), the
10
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 co,
we find da/drj cos co by orthogonal projection of AB on AC. Then
the division by cos (a co) furnishes AC and a second division
by cos (a co) leads to AC. Thus a certain curve can be
constructed whose polar coordinates are r = dfjdy and co, the
equation in polar coordinates being
da
COS CO
cos 2 (a co)
or
r / \v>
[r cos (a co)] r cos co.
drj
In rectangular coordinates , rj the equation assumes the form
(cos a + sin m?) 2 = -- .
cfy
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.
A G is a tangent of the parabola. Bisecting AW in E, drawing EK
perpendicular to AB f 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 ~, , this line would be independent of the unit of
length. For if I is the line representing the unit of length and
/ , I" the lines representing the values df/dy and ,, , df/dy
would be the ratio l /l and TTTT- the ratio I" /I; hence I" =>// .
Since V 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 ,, 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 will show itself, when 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, yi).
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 + Jc 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 fa, fe, k s , h by the following equations
DIFFEKENTIAL AND INTEGRAL CALCULUS.
ki = f(x, y)h,
133
4 = /(* + , y +
We then form the arithmetical means
+ k 3
. KI ~\- #4
and Q = ~ >
and find with a high degree of approximation as long as h is
not too large
k = P + l(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 Mathematik 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 = x , x = x\, x = x 2 , x = x s ,
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), f(xi, y}, f(x 2 , y),
Starting from a point A(XQ, yo) on the first vertical x = X Q (Fig.
91) we proceed to a point BI 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 /(XQ, 7/0). 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 ?/-axis and PO is equal to the unit of length by which the
lines representing f(x, y) are measured. Thus
OA"/PO = /(*, yo).
Now we draw AB\ perpendicular to PA", so that if h and 1
denote the differences of the coordinates of A and B, we have
h/h = OA"IPO,
h = /(.TO, y Q )h.
From Ci the point of intersection of the line AB i and the vertical
x = Xi 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 fe we have
or
From <?2 the point of intersection of the line AB% and the
vertical x = Xi we find in the same way a point Ba on the vertical
DIFFERENTIAL AND INTEGRAL CALCULUS. 135
x = x 2 and the difference fe between the ordinate of #3 and that
of A is
=/ ( *o + g #o "^~ 2/
From #3 we pass horizontally to IV on the curve representing
f(x 2 y) and vertically down to $ 3 ". The line AB^ is then drawn
perpendicular to PB^", so that the difference k between the
ordinates of B and A is
The bisection of BzBs and of B\B gives us the points EI and EZ
and the point B is taken between EI and E^, so that its distance
from EI is half its distance from E%. 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 s , 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 GKAPHICAL 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
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
d~x = isa ^ =COSa
Consequently
or
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, tga).
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.
ffiy
1
da
1
da
dx 2
cos 2 a
dx
cos 3 a
d8 9
da
~ds =
cos 3 a
dx 2
DIFFERENTIAL AND INTEGRAL 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 Mb at a distance from B equal
to the value of p that corresponds to
B and to the direction in which the FlG> 92 .
circular arc reaches B. With Mb as
center and MbB 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 GKAPHICAL 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
MbB, 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
dy
may be written in the form of two simultaneous equations of the
first order:
*-
dx~ Z
dz
-*(*,*,).
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:
DIFFERENTIAL AND INTEGRAL CALCULUS. 139
dz
Tx -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, z)
and g(x, y, z) 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
-JVf-
FIQ. 93.
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, z)
and g(x, y, z) are plotted as ordinates to the abscissa x, but for
clearness sake not in the same system of coordinates as y and z,
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 P\(x p , y p ) and P 2 (x p , z p ) in the first system
and the values of f(x p , y p , z p ) and g(x p , y p , z p ) by the two points
AI and A 2 in the second system of coordinates (Fig. 93). The
points AI and A 2 determine certain directions MA\, and MA 2
of the curves x, y and x, z, the point M (Fig. 93) being placed at a
distance from the axis of ordinates equal to the unit of length by
which the ordinates representing f(x, y, z) and g(x, y, z) are
measured. Through PI and P 2 we draw parallels to MA\ and
MA 2 as far as Qi and Q 2 with the coordinates x q , y q and x q , z g .
With these coordinates the values f(x q , y q , z q ) and g(x q , y q) z q )
are determined, which we represent by the ordinates of the
points Bi r B 2 . These points again determine certain directions
parallel to which the lines QiRi and Q 2 R 2 are drawn, etc. In this
manner we find first approximations y\ and zi for the functions
y and z and corresponding to these approximations we find
curves representing /(#, z/i, z\) and g(x, y\, Zi). These curves are
now integrated graphically, the integral curve of f(x, 2/1, zi)
beginning at PI and the integral curve of g(x, 2/1, Zi) at P 2 and
lead to second approximations y 2 and z 2 :
2/2 = y p + I f(x, 2/1, zi)dx,
j Xp
zz = z p + I g(z, yi, zi)dx.
Jr P
For these second approximations the values of f(x, y 2 , 22) and
g(x, y 2 , %) are determined at a number of points along the curves
x, y 2 and x, z 2 sufficiently close to construct the curves representing
f(x, 2/2, 22) and g(x, 2/2, 2 2 ). By their integration a third approxi
mation 2/3, 2 3 is obtained
I /(*
J *P
2/3 = 2/P
= z
I g(x, 2/2,
JX P
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
&-* V* y <fc,
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+ i = y p + I f(x, y n , z n )dx; z n+ i = z p + I g(x, y n , z n )dx.
J*p Jx p
For the true curve that passes through the point x p , y p , z p we
find by integration
y
hence
r* r x
= yp + I /fo 2/> *)*; 2 = *p + I g(*> y, *)<&;
Jx p Jfp
yn+i - y = I [f(x, y ny z n ) - f(x, y, z)]dx;
J x p
z n+ i z = I [g(x, y n , Zn) g(x, y, z)]dx.
JX P
142 GRAPHICAL METHODS.
Now let us write
f x f f x f(*> Vn, Z n ) ~ /O, y, Z n )
f(x, y, z n ) - f(x, y, z) = - _ - (y - y)
, /(a;, y, z n ) - /fo y, 2) ,
+ ~ -- - -- (Zn
z n z
and similarly
The quotients of differences
/O, y n , z n ) - f(x, y,
g(x, y, z n ) - g(x, y, z)
y n - y
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 df/dz, is not greater than M\, and that of dg/dy and
dg/dz not greater than M%, 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
Mi(8 n -j- e n ) and M 2 (S n + e n ).
Hence for the maximum values of y n+ i y and z n+ i z, which
are denoted by 5 n+ i and e n+ i we obtain the limits
5 n+ i ^ Mi(d n + ) | x x p | , n +i ^ M 2 (d n + e n ) \x x p \,
and
x - x p | (5 n + 6 n ).
If therefore the interval x or p of the integration is so far
reduced that
DIFFERENTIAL AND INTEGRAL CALCULUS. 143
5n+i + en+i is not larger than the fraction k of (5 n + c n ), but
from the same reason
( + e n ) ^ &(5 n _i + n _i), (5 n _i + Cn-l) < &(6n-2 + Cn- 2 ), etc. J
therefore
That is to say, for a sufficiently large value of n 8 n +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.
When 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
fa 1 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 , y r , 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
dy . dz ,
Tx =f(x,y,z), Tx = g(x,y,z).
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 h of x by the fol
lowing formulas analogous to those used for one differential
equation of the first order:
h = f(x, V, z)h; /i = g(x, y, z)h;
A .* j> Ji\i i ( . * > . Ai
= /I ar+ ^ , y+ TT > *+ 2 J 2= g \ x ^~ 2 y ~*~ "2 *~*~ 2 J ;
. . /
2 y+ ^ S+ 2J
= /(a; + A, H- fe, 2 + k)h; k = ^ + h, y + h, 2 + A;
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 + h/2, x + h. In the next
step where x + h, y + k, z + / 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 -}- %h, 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, xz, 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 0) xi, x z ,
For each of these constants / and g are functions of two inde
pendent variables and as such may be represented graphically
DIFFEKENTIAL 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 =
x =
x =
etc.
On the first x = XQ we mark two points with ordinates yo and ZQ,
and from the drawing that gives the values of /(XQ, y, z) and
FIG. 94.
g(xo, y, z) as functions of y and z we read the values /(XQ, yo, ZQ)
and g(xo, yo, z<j) and draw the lines from XQ, yo, and XQ, ZQ to the
points
afe, 2/o + h and x 2 , Z Q + h.
The intersections of these lines with the parallel x = x\ furnishes
the points
11
146 GKAPHICAL METHODS.
.fa . ^1
xi, 2/o + -^ and xi, ZQ + - .
With these ordinates we find from the second drawing the values
and by their help we can draw the lines from XQ, 2/0 and XQ, ZQ
to the points
2, 2/o + fa and x 2 , 2/0 + k.
The intersections of these lines with the line x = Xi furnishes the
points
, fa . k
xi, yo + ^ and x lf z + - ,
and with these ordinates we find the values
, ^2\ / ,fe ,/2\
, ^oH-^ J> 9( x * 2/o+*y,s + ^ J>
which enable us to draw the lines from XQ, 3/0 and XQ, ZQ to #2,
?/o + fe and x 2 , z + / 3 .
With these two ordinates we find from the third diagram (x = x%)
the values
f(x*, 2/o + fa, ZG + k) and gfa, y + fa, Z Q + / 3 ),
which finally enable us to draw the lines from #o2/o and X O ZQ to
22, 2/o + fa and z 2 , 2 + h-
On the vertical line x = #2 we thus obtain four points, BI, BZ,
#3, #4, corresponding to 2/0 + fa, 2/o + fe, 2/o + fe, 2/o + fa and
four points, BI, BI , BJ, B 4 f , corresponding to ZQ + /i, z + h,
20 + 4, ZQ + I, (Fig. 94).
$2# 3 and BiBi are bisected by the points Ci and ft; Bz Bs
and ft ft by the points Ci , ft . Finally ftft and ft ft are
divided into three equal parts and the points B and B are found
in the dividing points nearest to ft and ft .
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 z found for x = a- 4 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
d n x ( dx dx n ~^
Cut \ dt dt
or in the form of n simultaneous equations of the first order
dx
dt
A more general and more symmetrical form is
dx
~fa = fl(t, X, Xi, X n -l)>
, xi, a?n-i),
x n -i
fa- = f(t, X, Xl, Z n _i).
The functions x, xi, x 2 , x n -i are then represented as ordinates
to the abscissa t, so that we have n different curves. When the
function f(t, x, Xi, X<L, x n -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\,
Xn-ij there is no difficulty in constructing n curves whose
ordinates represent the functions x, Xi, x 2 , x n -i. Starting
from given values of t, x, x\, x%, x n -i we have only to apply
the same methods that have been explained for the first and the
second order.
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