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I. F. Quinby Horatio Nelson Robinson.

A new treatise on the elements of the differential and integral calculus

. (page 11 of 22)

208 DIFFERENTIAL CALCULUS.

point of which x, y, 2, aro the co-ordinates : then, if x, y, z,
are the same in the three Eqs. 1, 2, 3, the concrete question is
reduced to the abstract one of finding the values of x, y, z ;
which, when subject to the conditions of Eqs. 2 and 3, will
render F{x, y, z) a minimum.
By Art. 128, we have

2x + ixiA + fi^A' = ^

^I/ + f^iB + fi,B^ = ^ (4).

Multiplying the first of Eqs. 4 by Aj the second by B, and the
third by (7, adding the results, then, by (1), we have

{A' + B^ +C')fi, + (AA^ + BB' + CC')(i^ + 2D = (5).
In like manner,

{A^' + B'^+C''')lii + {AA'+BB' + CC')iJi^+2D' = .(6).
From Eqs. 5 and 6, we get the values of jWj, fi^'^ and Eqs. 4,
when these values of fi^ fi2, are substituted in them, will de-
termine x, y, z. Multiplying the first of (4) by x, the second
by y, and the third by Â«, adding results, and reducing by Eqs.
1, 2, and 3, we have

2F{x,y,z) + Dii,+iyfi, = 0^,

from which we get F{Xj y, z). In this case, it is unnecessary
to examine the sign of F{x -j- A, y + fc, c + **) â€” F{x, y, z),
when the values of Xj y, z, are substituted; for we know from
the conditions of the geometrical question that the function
has a minimum.

7. Required the values of x, y, z, that will render the func-
tion

u = x^y^z^

a maximum, the variables being subject to the condition
t;=:aa? + 6y + cÂ« â€” i = 0.

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EXAMPLES, ^MAXIMA AND MINIMA. 209

We find

dx -^ ^ X ' dy y dz z '

dv __ dv r ^^

dx"^ ^ dy ^ dz

therefore (Art. 128)

p q ^_i> + g' + ^

ax hy cz A '

p h ah r h

^p-{-^ + ^ ^i^ + ffH-^' cjp + g + r'

These values of x, y, 2, make the function a maximum. For
we find

d'^u p du p d^u q du q d'U r du r

dx^'^xdx x^. [ o[y^ ^y 'Sy'^y^ ' dz^ '^ z dz z^

and these, because -, = 0, -j- = 0, -t- = 0, for the values of

Xj y, 2, become

d'^u _^ p ^^^^ _ q d'^u ^ r
d^^^x^^'d^^^^y'^'W'V^^'

all of which are negative, â€” a necessary condition for a maxi-

d u d u d u
mum; and, by getting the partial derivatives^-,-, , , , yâ€”rj

we see that the other conditions (Art. 126) to insure this state
of the function are also satisfied.

By making aâ€” 1,6=1, c = l, the above becomes t!ie solu-
tion of the problem for dividing the number k into three such
parts, that the product of the p power of the first, the q power
of the second, and the r power of the third, shall be a maxi-
mum.

8, Inscribe in a sphere the greatest parallelopipedon.

f If a be the radius of the sphere, the parallelo-

Ans. i . , . . 2a .

pipedon IS a cube having â€” ^ for its edge.

27

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210 DIFFERENTIAL CALCULUS,

9. Determine a point within a triangle, from which, if lines
be drawn to the vertices of the angles, the Bum of their
squares shall be a maximum.

C The point is the intersection of the lines

Ans. \ drawn from the vertices of the angles to

Lthe centres of the opposite sides.

A function F{Xj y, . . .) of two or more variables may be of

such form, that it admits of a maximum or minimum for values

of the variables which make -7-, -,-,... indeterminate or in-

dx ay

finite. Tliere are no general rules applicable to such cases;

but each one must be specially examined.

1 0. What values of x and y Avill make

u = ax^ + (x^ + by'^)i
a minimum?

du ^ ,2 x du 2 y

dx ^ 3 (a;2 + lyif dy 3 (a;2 + hy'^f

For rr = 0, y = 0, these difierential co-efficients take the form
- : but their true values are infinity ; for, if we make y =z mXj

they become

die ^ 2 1 die 2 m

= 2ax + ^-

dx '^^x^{l+m''bf dy ^xi{l+m''b)^*

Hence for a? = 0, and therefore for y = 0, at the same time,
we have

du ___ du _

rfx ~" * dy'"
For a; = 0, y = 0, we have w =1 ; and no real values of x and
y can make u negative. Hence w is a minimum for x = Oj

y = 0.

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SECTION XIII.

CHANGE OP INDEPENDENT VARIABLES IN DIFFERENTIATION.

129* It is often required in investigations to change dif-
ferential expressions, obtained under the supposition that cer-
tain variables were independent, into their equivalents when
such variables are themselves functions of others.

Suppose that, having given y=if[z),x=iq>(^z), it is required
to express the successive derivatives of y, taken as a function
a?, in terms of those of x and y taken with respect to z.

We have found (Art. 42)

dy dy dz

dx dz dx'

and (Art.

41)

dy

dz _

1

dy

dz

dx ~

= dx'- â€¢'
dz

' dx-
dy

dx '
di

dhj
dx^

d

dz d

dz dz

Hence

dx dx dz

dx dx

(Art,

42)

dz

dz

d - y

dx d"x dy

'dz''

dz dz'

' di dz

(Â£)â–

3i

d'y

dx d'^x dy

_~dz'

dz ~ dz'

'â€¢ dz .

dz

1

/dxV
[dzj

' ^'"""^ 35

Tz

211

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212 DIFFERENTIAL CALCULUS.

So also

d^y dx <Px dy
d^y _ d dz^ ~dz ~" dz^ dz ^_â€ž_ -

dx' "" dx W^^Y
W

d^y dx d^x dy
_ ddz^ dz^'dz^ dz dz
"^ dz /dxy dx

\dz)

Yd^y dx _ d^ dy\ (dx\* _ /dxV d^x /d^y dx _ d*x dy\
_ \crz^ dz dT' ~dz)\dz) \dz) d? \^' dz dz" dz/ dz
~ /dx\^ dx

/d*y dx d^x dy\dx d^x/d^ydx d*xdy\
\d^' dz~d?"dz)dz~ dz^\d? dz~dz^Tz)

- 7^Y *

\dz)

d^v d^ y
In the same manner, we may find ^ H zr a* â€¢ â€¢ Substitut-

dx u7j d X
ing in these the values of -r^-, -,^, .-^ â€¢â€¢â€¢, found from

we have the values of the successive derivatives of y
with respect to x, in terms of those of x and y with respect
to 2.

130. Having y =z/(x), to change the independent variable

from X to y in the expressions for -,^ , -^-^ . . .

dx dx^

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CHANGE OF INDEPENDENT VARIABLE. 213

dy

^^y â€” A .L ~ _^ -L % /Art. 42)
^^ cZx dx dy dx dx ^ '

d'^x d'^x

_ dy^ dy ___ dy^

Similarly,

dxV dx "^ /dx\^'

dy) \dy)

d^ d^

^ â€” __ _1 dy^ â€” _ iL -^^ ^y

dx' "^ da; /dicy "" *" dy /dirV di

fjo/dxV _ /dxV/d^xV

^ dy'\dy) \dy)\dr)

/dx\^

\dy)

d^dx __ fd^V
z= _ dy^ dy "~ \dy^/

/dxV

\dy)

In like manner, we may find the expressions for ^â€” ^, -j\*"

These formulas may also be found from those in the preceding
article, by making 2 = y ; whence

dy_^ d^y __^ ^^_a dx _dx d''-x _^d'^x

dz"" ' d22 - "> dz^^^''-' dz^dif d2~^ ""dp â€¢â€¢â€¢
By the introduction of these values in the formulas of Art

129, they will be found to agree with those just established.
J3 J. Having given y=f(x) (1),

and also x:=it cos. 6, y :=ir sin. (2),

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214 DIFFERENTIAL CALCULUS,

it is evident that we may eliminate x and y from these equa-
tions, and get a direct relation between r and 0; and thus r
becomes a function of 0,

It is required to express the values of ,-> ^-,1 â€¢ â€¢ â€¢ ; derived

dv d^7*
from Eq. 1, in terms of -^y -z-^* â€¢â€¢
^ ' do dO^

By Arts. 41, 42, we have

^1
dy ^dy do _dy 1 __ dd

dx do dx do dx ~~ dx

do do

dv
sin. - , - {' r cos.
do '

~^- from Eqs. 2 ;

COS. 6 ' '^

also

COS. J- â€” r sin.
do

dv dy

djj^d Bin.^^- +rcos.^ ^ sin^gg^-frcos.^ ^^

dx' dx j^ . , "^^ ^dr . dx

COS. ^ -,- â€” r sm. cos. ^^ â€” r sin. ^

do do

Performing the indicated differentiation, we find for the nu-
merator of the result

I , d^-r ^ ^ dr . \/ dr . \

( sm. -â€” - + 2cos. r sm. 6 jf cos. 0~ r sm. d ]

\ do' ^ do )\ do I

I d'^r ^ , dr \ / , dr \

â€” - COS. â€”- â€” 2sm. ^- rcos. ^ ) sm. -â€” + ^cos.^ ,

V do' do )\ dO^ /'

/dr\^
which reduces to r^ + 2 ( -7- ) â€” r

^ \d0j

/dr\^ d''r^

"do''

dx dr

and the denominator, remembering that -3- = cos, ^3 rsm.^,

aO do

IS

/ dr . y

[ COS. â€” r sm. )

\ do J

Digitized by

CHANGE OF INDEPENDENT VARIABLE.

215

Hence

COS. 6 r sin. ^

dJ

These formulas are used in the applications of the differen-
tial calculus to geometry, where a change of reference is
made from rectilinear to polar co-ordinates.

132. Suppose that we have the expressions for - ; â€” ,

CLx ay

found from the equation u = F{x, y) ; but that the variables

X and y are connected with two other variables, r and 6, by the

equations a? =: jp\(r, ^), y ^= F^iTjO): then we may conceive

X and y to be eliminated from these three equations, and u

to be a function of r and 0, Required the equivalents of

the expressions for ,- , - - , in terms of the derivatives of

X, y, and Uj Avith respect to r and 0,
By Art. 82, we have

du du dx du dy

dr dx dr dy dr

du __dii dx du dy
do "" ~dx ~dO ' dy dO

(1);

and from these two equations the values of

du du
dx' dy'*

can be

found.

When the equations expressing the relations between
Xy y, r, e, are r =: Fi{x, y), d=,F2{x, y), instead of those
given above, then

du _^du dr , du dd

dx~~

du
dy'

dr dx do dx

du dr ^.du dd
dr dy"^ dd dy

(2).

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216 DIFFERENTIAL CALCULUS.

If the variables x, y, r, 0, are connected by the unresolved
equations Fi{x, y, r, 0) ==. 0, Fo(x, y, r, C) == 0, we proceed
thus : â€”
By Art. 82,

dFA dF^ dx dF^ dy _

Q

^rfx d>^rfy dd ^'

(d^\.dI\dxdF\dy^
\do)'^dx dd'^dy dd '

in which it must be remembered that ( â€” ^ j , I â€” â€” ? j, are par-
tial derivatives of i^i, F2, with respect to 6,

Differentiating F^, F^, with respect to r, we get two similar

dx dv
equations involving ^-, -^ , and the four equations thus ob-
tained will determine -,-, -^-, ^, -,-, which must be sub-
dd do dr dr'

stituted in formulas (1) or (2), Art. 131.

The following example will illustrate the manner of using
the above formulas : â€”

Given u =/{Xj y), x = r cos. 0, y =ir sin. 6, it is required

du du , ^ /. ^ du du

to express ^^, ^^, m terms of r, 0,-^^, ^.

We have

dx ' r. dy

,- = â€” r sm. 0, -/- = r cos. d,
do 'do '

dx . dy . ^

-i- = cos.^, -^- =1 sin. 6,

dr dr

Hence, by formulas (1),

du ^dti , , ^du '

â€” - = cos. ^ T- + sm. -f~ ,
dr dx dy

du , ^du , ^du

-.-=1 ^r sm. -i- + r cos.^ ^p- :
dd dx dy

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CHANGE OF INDEPENDENT VARIABLE. 217

(a).

- da ^du \ , ^du

whence -^ = cos. 0-^ sm. -^-

dx dr r dd

du . ^du , 1 ^du
-^ = sm. 0-^ â€” â€” COS. d -7-
dy dr r dd

To make formulas (2) applicable to this example, we first
deduce, from the equations x=ir cos. 0,y :=r sin. dj the values
of r and in terms of x and y. We find

r = Vx^ + y\ = tan.-i ^ ;

X

, dr X dr y dO y dO x

whence :t-=-, -t =~, zr = â€” 2' zr = -2'
dx r dy r dx r^ dy r*

and, by means of these, Eqs. 2 become

du __x du y du^

dx~^ r dr r^ d

du __y du X du
dy r dr r^ dO

X

The relations x =z r cos. 6, y = r sin. 6, give cos. ^ = -,

sin. ^ = - , by means of which we can pass from formulas (6)
to (a), or the opposite.

133* Attention is here called to the necessity of attaching
their precise signification to the symbols

dr dr dx dy dd dd dx dy
dx' dy' ~dr' rfr' dx' 'dy' W W

which occur in formulas (1) and (2), Art. 131.

It must be borne in mind that these denote partial differ-
ential co-efl5cients, and that those referring to the same varia-

dr dx
bles, such as -^- , -^ , have not to each other the relation of
CLx dr

dii dx

^ to ,-, which are derived from the equation /(x, y) = 0.

Cm) dy

With reference to these last, we know that one is the recipro-

23

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218 DIFFERENTIAL CALCULUS.

cal of the other, or that their product is 1 ; but this is not

dt* dx
true for â€¢ X , â€¢ The consideratiou of the meaning of the

term " differential co-efBcient," and the difference between the
equations connecting the variables in the two cases, will re-
move all difficulty. In getting formulas (1), ic and y were given
as explicit functions of the independent variables r and ; and
a change in either r or will produce changes in both x and y.

dx
Hence, in the operation of finding -r- , r, Xy and y vary, while

d remains constant. In formulas (2), r and were given as
explicit functions of x and y ; and a change in the value of
either x or y will produce changes in both r and ; and hence

dr

the increment attributed to x in getting -p causes r and

also to vary, while y remains constant. That is, in formulas

dx

(1), -J- supposes r, x, and y to vary together, being constant;

dj*
while, in formulas (2), -^ supposes a?, r, and to vary, while y

remains constant. Thus it appears that these two partial de-
rivatives are obtained on different suppositions in respect to
the variables which receive increments, and those which
remain constant.

In the example just given for formulas (1), we have

(Ix dv

â– :.- z=z cos.^; and, for formulas (2), -^- = cos. 6 ; and the product

dx dr ^^

, -.â€” =1 cosrO.
dr dx

134. Having u = F(Xj y, z), and three equations express-
ing the relations between Xj y, 2, and three other variables

7% Of ipj it is required to find the values of -7- , -,- , -y- , in
terms of the different co-efficients of u witli respect to r> 6^ xp.

Digitized by

GoogJe

CHANGE OF INDEPENDENT VARIABLE,

219

By Art. 82,

du
dx

du do j^du c?V _, du dr

do dx dip dx dr dx

du ^du do du dip du dr

dy dOdy dxp dy dr dy

du

dz '

du dd du dip du dr
do dz"^ dip dz "^ dr dz

(1).

The three equations connecting x, y, z, r, 0, xp, Avill enable

^ , , . do dd do dr dw i t. ^ i_

us to determine , , -y-j -, > T ' ' ' ' ;/ â€¢ â€¢ â€¢ > ^^^ ^^^' ^i when

these values are substituted in them, give us the expressions
sought.

By solving Eqs. 1, wo can also find the values of

du du du
do' W d^'

J . ^ ndu du du /? 1 ^t

expressed m terras oi ,- , -r- , -^^ ; or we may find these

values from the equations

dz

du du dx du dy du dz

'dO'^dx 'dO^dy 'do'^'dz dO

du du dx du dy du dz
dip dx dip dy dw dz dip

> (2).

du du dx du dy du dz
dr dx dr dy dr dz dr

135. Let the relations between the variables x, y, z, 0, ip, r,
be

x = r sin. cos. ip, y z=.r sin. sin. xp^ z = r cos. (V),

From these we find

dx ^ dy ^ . dz . ^

^ zz: r COS. Â§ COS. \py ^^ = ^ COS. 6 Bm. \p, ~j :=i â€” r sm. ^,

dx â€¢ z) â€¢ dy ' ^ dz ^

-T- = â€”rsm.^sm.t^, -^- = r sin. <? cos. t/;, -r- = 0,

dx . . dy . ^ , dz

T- = sm. cos. Tp, ;r = Â®^^' ^ ^^^- V*? j" = ^o^ ^ /

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220

DIFFERENTIAL CALCULUS,

and formulas (2), Art. 134, by the substitution of these values,
become

du ^ du . ^ , du â€¢ ^du

-â€” =.rcos,Ocos,xiJ â€” -{-rcos.dsm.yj â€” â€” rsin.^ â€”
do dx dy dz

du ' n ' du , , ^ du

-â€” = â€” r sm. 6 sin. \p â€” -\- r sin. cos. w â€” l (a).

dip dx ' dy ^

du . ^ du . , ^ , du . ^du

_.- = Bin.Ocos.yj - -. -f- sin.^sm. i/i â€” - + cos. <!^ - -
dr dx dy dz

From Eqs. a may be found the required values of -r- , -,â€” , -r- ,

in terms of

du du du

dd' dip' dr'

Again : squaring Eqs. 1', adding results, and taking square
root, we have r =: \^{x'^ + y^ + ^^)- Adding the squares of
first and second of these equations, we find r ^ sin.^ = x^-{-y^]

whence r sin. d = V{^^ + 2^^)? ^^^- ^ = - V(^^ + y^)' and from
this, and the last of Eqs. 1', wo find

tan.(?=:!LJ^ â€” Z_:iJ ^ = tan.-^^l-^ â€” XfJl.
z z

Dividing the second of Eqs. 1' by the first, we have

tan.t/;=: ?^, t/;z= tan."^^.
x x

Hence we have

r =z ^{x' + y' + 2^), =z tan.-^ ^^' + ^ - , ^P = tan.-^ I (20-

z X

From those by differentiation, we have

dr X . ^ dr y . ^ . dr z

, z=z- = sin. COS. xp, - = ^- =: sm. sm. v^ -r =~ = cos.O,
dx r dy r ^ ' dz r

do

X

COS. COS. xif

c?^ z . y COS. d sin. \p

. X

rfy"~aj2+y2 + 2'^^ Vi' + y3

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CHANGE OF INDEPENDENT VARIABLE.

221

^ â€” _ V(^' + y^) _ _ sin. 6

dip y _ sin. ip d\p __ X

dx ic^ + y*-* rsin. ^' rfy "~ a;^ +y*^

By the substitution of these values in formulas (1), Art. 134,
we have

C0S.1/; dip ^

/2 r sin. 0^ dz

dti COS. d COS. 1/; du
dx r do

Sin. ti; du ^ , ^ du

: -, 1- Sm. COS. \i) -y-

r sin. dip ^ ^ dr

du COS. d sin. t/; c?i^

du

dz

- -. COS. t/; du
dJ ^ r sin. ^ rfv'

r

sin. (9 c?i^ rfi^

-^ do -^ '''''' ^d?

du
4- sm. (9 sm. xp -p

(6).

The values -5-, -,-, -y-, given by formulas (a), will be found
to agree with those given directly by formulas (6).

EXAMPLES.

1. Transform

d^y /dy\* dy

dx

dx

(1)

into its equivalent when neither x nor y is independent, but

both are functions of a third variable z,

d^ u dii

Substitute for -^ and ^- their values given in Art. 129,

and we have

d'^y dx d^x dy
dz'^ dz ~dz'^ dz

/dxV
\dz)

fdx\
and, multiplying through by / -_- j

X

d-y dx d'^x dy /dyV dy /dx\}

dz^ dz

â€¢ X -4- I â€”

dz^ dz \dz

d.\r.r' ('>â€¢

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222 DIFFERENTIAL CALCULUS.

If we make a; = z, this reduces to the given equation.
Making y = 2, (2) becomes

Equation (3) is the equivalent of (1) when the independent
variable is changed from x to y.
2. Change

_ X fdx\^

i(l)'+f+Â«0=Â»

into its equivalent when both x and y are functions of a third
variable 2. "

If y ^ z, the above becomes

in which y is the independent variable.
3. Eliminate x between

and find what the diflferential equation is when 6 is the inde-
pendent variable, and also when y is the independent variable.
First suppose both x and y to be functions of a third varia-
ble, z; then the diflferential equation becomes (Art 129)

dz^ dz dz' dz'^ xdz \dz) "^ ^ \dz) "" ^^*
From a;2 = 4/9, we have x = 2^^, -^- =-17.-^-' ^^^:j7 = ^^^J

^ic?^ rf^ic d idd ^^id^d 1 t/M^

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CHANGE OF INDEPENDENT VARJABLE. 223
In (2) replacing x, ,- , ; j; by their values, we Lave

which does not contain x. Making if =^z, (3) becomes
and if, instead, =zZj we have

4. Given the relation a; = e% to change the independent
variable, in the differential expression a?'* ^,7, from x to s.
By Art. 42, we have ,

or, writing the first member in an abbreviated form,
fd \ d'y â€ž.,c^" + 'y /iv
\d8 ) dx'^ dx" + ' ^ '

Making n = \, this gives

^-lV^^=^^^ (2).

c& /tie rfic^

dx
From re â€” e*, we get -^ =ze* = x; also we havÂ©

d2/ dy dx dy ^

ds ~~ dx ds ^^ dx

hence (2) becomes

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224 DIFFERENTIAL CALCULUS.

When 71 ^= 2 in formula (1), then

a-^>

dx'^ dx

d^ V
and, putting in this the value of x^ -â€” from (3),

dx' \d8 J\ds Jds

The law governing the construction of these equations is ob-
vious; and wo may write, generally,

The meaning of the operations denoted in the second member
of formula (4) is, that if the expressions -^ 1, -^ 2 . . , ,

CL8 CIS

bo combined by the rules for multiplication, the result will
represent, in terms of indicated diflFerentiations on -^-, the value

of aj"-T-?.

dx*"

h+mV

\dx/ \
5. If we have p = ^ -^ ^ , and the relations

dx^
x = r cos.Oj y =zr sin, dy find the equivalent for p when a
change of independent variable is made from x to ^, and also
from X to r.

When d is the independent variable.

r

tWA'

dd^ \dd)

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CHANGE OF INDEPENDENT VARIABLE, 225

and, when r is independent,

_j

P=â€” â€”

d^O dd ^dd*

rfr^ ^ dr^ dr

(

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SECTION XIV.

ELIMINATION OP CONSTANTS AND ARBITRARY FUNCTIONS BY
DIFFERENTIATION.

136* When an equation is given in the form

F{x,y) = c (1),

the constant c will disappear on the first differentiation , and
the successive differential equations derived from (1) will be
identical with those derived from

Fix,y) = (2).

Though an equation may not be given under the form of (1),
it often happens that one or more of its constants may be made
to disappear by successive differentiation alone.

Let {ij - by + (x â€” a)2 - r^ = (3),

and differentiate this equation twice, taking x as the independ-
ent variable. We find

{y-b)^ + x-a = (4),

(^-^)S + ('SY+1 = (5);

and thus the two constants a and r of (3) have vanished in
the two differentiations which lead to (5). A third differen-
tiation gives

dx^ dx dx'^

226

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ELIMINATION OF CONSTANTS AND FUNCTIONS. 227

From Eqs. 3, 4, 5, and 6, we get

dy^x-^a d'^y _ (x ^ af + (b -^ yf __ r^

dx b â€” y dx' (6 - yf {b â€” yf

d^jy ^ _ ^dx_dxl _ ^r \x - a )
dx' y-b - {b-yf'^

and, by eliminating y â€” 6 between (5) and (6), we get

l(D'+'!0-l(2;)'=Â« <')â€¢

Between (3) and (4), we may eliminate any one of three con-
stants a, b, r; and, by taking these constants in succession, we
should have for our results three diflFerential equations of the
first order, each containing two of the constants. By a proper
combination of (3), (4), and (5), wc can arrive at two differen-
tial equations of the second order, each containing but one of
the constants of the primitive equation; and between (3), (4),
(5), and (6), we can eliminate all three of the constants, ob-
taining for the result a single differential equation of the third
order.

It thus appears, that, by differentiation and elimination, Eq.
3 will give rise, 1st, To three differential equations of the
first order, each involving two of the constants a, b,r ; 2d, To
two differential equations of the second order, each involving
but one of these constants; 3d, To one differential equation of
the third order, from which all of the constants have vanished.

By means of Eqs. 3, 4, 5, the values of a, b, r, may be ex-
pressed in terms of x^ y^ and the derivatives of y of the first
and second orders. Denoting these derivatives by y^, y^^j wo
find

^ â€¢ y" ^ yf ' -*- y"

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228 DIFFERENTIAL CALCULUS.

137* III general, if we have an equation between x and y,
and n arbitrary constants, and we differentiate this equation
m times successively, we shall have, with the primitive equa-
tion, m -\-l equations, between which we can eliminate m
constants. This will lead to a differential equation of the m}^
order, in Avhich there will be but n â€” m of the constants ; and,
as the constants eliminated may be selected at pleasure, it is
evident that as many equations of the order m may be formed,
each containing n â€” m constants, as we can form combinations
of n things taken m in a set, which is expressed by

1.2.3. ..w

When the original equation is differentiated n times, we
should have altogether n -j- 1 equations, between which the n
constants can be eliminated; and, as the resulting equation
would involve the n^^ differential co-efficient of y taken with
respect to x, it is said to be of the n^^ order. The order of
the highest differential co-efficient entering any of the equa-
tions at which we arrive, by the steps above indicated, deter-
mines the order of the differential equation.

It is worthy of remark, that if any one of the differential
equations of the rn}^ order, obtained by eliminating between
the first m derived equations, and the primitive equation, m of
the constants entering the latter, be differentiated n â€” m times
in succession, then this equation of the m^^ order, and its
n â€” m derived equations, would enable us to eliminate the re-
maining constants ; and the final equation at which we should
arrive would be the same as that obtained by effecting the
elimination between the primitive equation and its n succes-
sive derived equations.

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ELIMINATION OF CONSTANTS AND FUNCTIONS. 229

y'^ + 1
To illustrate, take the equation a = x â€” - â€” ^J^â€” , and diflPer-

entiate with respect to x. We should find, after reduction,

which agrees with Eq. 7.

The theory of the elimination of constants by diflPerentiation
is suflSciently simple, and needs but little explanation. It is
referred to here for the reason that a knowledge of the forma-
tion of differential equations assists in understanding the more .
diflScult and highly important operation of passing back from
such equations to those from which it may be presumed that
they have been derived.

138. Functions known and arbitrary may also be elimi-
nated by differentiation.

Let y = a sin. x ; then -^ =.a cos. x = Va^ â€” y^ :

.â–  + (^'-Â». = o.

an equation which no longer contains the known function
sin. x.

Again: suppose 2z=g)(- j, in which x and y are independ-
ent, and (p denotes a function of the ratio of these variables,
the form of which is not given, and is therefore called an arbi-
trary/unction.

Make t = ^] thena = (3p(0, ^=9'(0^=-9'W,
dz .... dt X . .

Â» dz . dz ^

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230 DIFFERENTIAL CALCULUS,

This last equation is true, whatever may be the form of the
function of - denoted by 9 ; it may be z=Zf-j, 2 = sin. -, or

z=zey : and for each of these cases the diflferential equation

subsists.

Take the more general case, w = g* (v), in which u and v are

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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