Page:On Faraday's Lines of Force.pdf/47

Rh then

$a'=a-\frac{dV}{dx}$,

$b'=b-\frac{dV}{dy}$,

$c'=c-\frac{dV}{dy}$,

satisfy the condition

$\frac{da'}{dx}+\frac{db'}{dy}+\frac{dc'}{dz}=0$;

and therefore we can ﬁnd three functions $$A, B, C$$, and from these $$\alpha, \beta, \gamma$$,. so as to satisfy the given equations.

THEOREM VII.

The integral throughout inﬁnity

$Q=\iiint (a_1\alpha_1+b_1\beta_1+c_1\gamma_1)dxdydz$

where $$a_1b_1c_1, \alpha_1, \beta_1, \gamma_1$$ are any functions whatsoever, is capable of transformation into

$Q=+\iiint\{4\pi p\rho_1-(\alpha_0a_2+\beta_0b_2+\gamma_0c_2)\}dxdydz$,

in which the quantities are found from the equations

$\frac{da_1}{dx}+\frac{db_1}{dy}+\frac{dc_1}{dz}+4\pi\rho_1=0$,

$\frac{d\alpha_1}{dx}+\frac{d\beta_1}{dy}+\frac{d\gamma_1}{dz}+4\pi\rho_1'=0$;

$$\alpha_0, \beta_0, \gamma_0 V$$ are determined from $$a_1b_1c_1$$ by the last theorem, so that

$$

$$a_2b_2c_2$$ are found $$\alpha_1, \beta_1, \gamma_1$$ by the equations

$a_2=\frac{d\beta_1}{dz}-\frac{d\gamma_1}{dy}$&c.,

and $$p$$ is found from the equation

$\frac{d^2p}{dx^2}+\frac{d^2}{dy^2}+\frac{d^2}{dz^2}+4\pi\rho_1'=0$.