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d? V d? V (5) Transform

= 0 into a function of p and

d x2 dy e, having given x = r cos 0, y = q sin 0.


dV sin

cos O dV +


dV cosO d’V cosA d V
sino e

dr p2 d
2 sin 0 cos e d'V dV

dr do




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dy The expression for

đV may be deduced from that of dy

dx by putting O for 0. We then get

da V

d? V sin d V sine Ꮎ d
= cos 0

d go's


7 dr
2 sin 0 cos ddV

dr de de
Adding these together,
đV đV dy

1 d V 1 dv
dx? dyi dr2

goz do

r dr
dv d V dy
(6) Transform



d x2 dy do2 into a function of r, 0, and , having given

X = r cos , y = r sin o sin , % = r sin 0 cos 0. A slight artifice will enable us to do this with considerable facility. Assume p=r sin e, so that

y = p sin , x = p cos ,
ps = p sin 0, Q = p cos .




= 0.


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Taking first the two variables y and %, we find as in the preceding example

d’V đV V 1 đV
dz dpi 'pi do

p2 dop de In exactly the same way, the equations of condition being similar, we find



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Also, as in the first part of the last example,


1 d V 1 DV cot d

p de
po dr

go2 d"
Adding these three expressions,

d'V div d’V

dy do d x2
đ? V
1 TV
1 d°2 dV cot d


dr2 po de
p dg po dr

22 d By substituting for p its value, and making some obvious reductions, this becomes

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(sino O d.cos e

= 0.

sin o do

This important equation is the basis of the Mathematical Theories of Attraction and Electricity. The artifice here used is given by Mr A. Smith in the Cambridge Mathematical Journal, Vol. 1. p. 122.

(7) Transform the double integral

Sfany"-1 dy dx into one where u and v are the independent variables, x, y, u, v being connected by the equations

W + y = U, Y = UV.

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Therefore dy dx = ududv, and Slam-lyu-1 dy dx = Sfum +--) (1 – v)m-\yn-Idu dv.

This transformation is given by Jacobi in Crelle's Journal, Vol. XI. p. 307: it is of great use in the investigation of the values of definite integrals.

(8) Transform the double integral

Sfex*+y* dx dy into one where r and are the independent variables, having given

X = r cos 0, y = r sin 0,
Sfero + yo dx dy = - Sférer dr de.

(9) Having given

20 = r cos 0, y = r sin o sin , * = r sin 0 cos 0, transform the triple integral

SSS V dy dx into a function of r, e, and p.

Using the same artifice as in Ex. 6, we find

SSSV dy dz = SSS V pol dr sin 0 de do. This is a very important transformation, being that from rectangular to polar co-ordinates in space. V = 1, SS du dy dz is the expression for the volume of any solid referred to rectangular co-ordinates: and it becomes SSS=2 dr sin 0 do do when referred to polar co-ordinates.

If we suppose

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(10) Having given z a function of x and y determined by the equation



a? 62 it is required to transform

dx Sidx dy {1 +


into a function of O and when

x = a sin 8 cos , y = b sino sin ,

and consequently x = c cos 0. In this case

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dx dy

dx dy

= ab sin 0 cos 0,
de' do do de
dx dy dx dy

- bc (sin 6)? cos ,
de' do do de
dx dx da dx

= a c (sin ) sin . de do do de Substituting these values in the general expressions for dr dz

and dx dy, we find dx' dy'

dz Ssdæ dy {1+

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= Ssdo do sin 0 {ab? (cos()? +(c sin )' (a’ sin’p + bocosop)}4.

Ivory, Phil. Trans. 1809. CHAPTER IV.



Ex. (1)

y' = ax + b.........(1).
To eliminate b, differentiate, when we have



= a.........(2).

dx To eliminate a, substitute its value given by (2) in (1);

dy then y* = 2xy

+ 6.

d x

To eliminate both a and b, differentiate (2) again ; then

dy dy y



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= 0.


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(4) Eliminate a and b from the equation

y - a mo? –

= 0;

dạy 2 dy 2y the result is


= 0. dixo

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