Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/37

Page 26, l. 3 from bottom, dele 'As we have made no assumption', &c. down to l. 7 of p. 27, 'the expression may then be written', and substitute as follows:—

Let us now suppose that the curves for which $a$ is constant form a series of closed curves surrounding the point on the surface for which $a$ has its minimum value, $a_{0}$, the last curve of the series, for which $a=a_{1}$, coinciding with the original closed curve s.

Let us also suppose that the curves for which $\beta$ is constant form a series of lines drawn from the point at which $a=a_{0}$ to the closed curve s, the first $\beta _{0}$, and the last, $\beta _{1}$, being identical

Integrating (8) by parts the first term with respect to a and the second with respect to $\beta$, the double integrals destroy each other. The line integral,

$\int _{\beta _{0}}^{\beta _{1}}(X{\frac {dx}{d\beta }})_{a=a_{0}}\ d\beta$,

is zero, because the curve $a=a_{0}$, is reduced to a point at which there is but one value of $X$ and of $x$.

The two line integrals,

$-\int _{a_{0}}^{a_{1}}(X{\frac {dx}{da}})\ _{\mathrm {\beta =\beta _{1}} }\ da$ + $\int _{a_{0}}^{a_{1}}(X{\frac {dx}{da}})\ _{\mathrm {\beta =\beta _{0}} }\ da$,

destroy each other, because the point $(a,\beta _{1})$ is identical with the point $(a,\beta _{0})$.

The expressions (8) is therefore reduced to

$\int _{\beta _{0}}^{\beta _{1}}(X{\frac {dx}{d\beta }})\ _{\mathrm {a=a_{1}} }\ d\beta$

(9)

Since the curve $a=a_{1}$ is identical with the closed curve s, we may write this expression

p. 80, in equations (3), (4), (6), (e), (17), (18), (19), (20), (21), (22), for $R$ read $N1$.

p. 82, l. 3, for $Rl$ read $N1$.

p. 83, in equations (28), (29), (30). (31), for $({\frac {d^{2}V_{1}}{dx^{3}}})$ read $({\frac {d^{2}V'}{dxdv}})$

p. 83,„ in equation (29), insert - before the second member.

p. 105, 1. 2, for $Q$ read $8\pi Q$.

p. 108, equation (1), for $\rho$ read $\rho '$.

p. 108,„ equation„ (2), for $\rho '$ read $\rho$.

p. 108,„ equation„ (3), for $\sigma$, read $\sigma '$.

p. 108,„ equation„ (4), for $\sigma '$ read $\sigma$.

p. 113, l. 4, for $KR$ read ${\frac {1}{4\pi }}KR$.

p. 113,„ l. 5, for $KRR'cos\epsilon$ read ${\frac {1}{4\pi }}KRR'cos\epsilon$.

p. 111, I. 5, for $S_{1}$ read $S$.

p. 124, last line, for $e_{1}+e_{1}$ read $e_{1}+e_{2}$.

p. 125, lines 3 and 4, transpose within and without; l. 16, for $v$ read $V$; and l. 18, for $V$ read $v$.

p. 128, lines 11, 10, 8 from bottom for $dx$ read $dz$.

p. 149, l. 24, for equpotential read equipotential.