A Treatise on Electricity and Magnetism/Errata

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

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

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

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

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

The two line integrals,

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

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

The expressions (8) is therefore reduced to

 ${\displaystyle \int _{\beta _{0}}^{\beta _{1}}(X{\frac {dx}{d\beta }})\ _{\mathrm {a=a_{1}} }\ d\beta }$ (9)

Since the curve ${\displaystyle a=a_{1}}$ 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 ${\displaystyle R}$ read ${\displaystyle N1}$.

p. 82, l. 3, for ${\displaystyle Rl}$ read ${\displaystyle N1}$.

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

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

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

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

p. 108, equation (2), for ${\displaystyle \rho '}$ read ${\displaystyle \rho }$.

p. 108, equation (3), for ${\displaystyle \sigma }$, read ${\displaystyle \sigma '}$.

p. 108, equation (4), for ${\displaystyle \sigma '}$ read ${\displaystyle \sigma }$.

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

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

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

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

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

p. 128, lines 11, 10, 8 from bottom for ${\displaystyle dx}$ read ${\displaystyle dz}$.

p. 149, l. 24, for equpotential read equipotential. p. 159, l. 3, for ${\displaystyle F}$ read ${\displaystyle f}$.

p. 159, l. 2 from bottom, for ${\displaystyle M}$ read ${\displaystyle M_{2}}$.

p. 163, l. 20, for ${\displaystyle \lambda _{i-s+1}}$ read ${\displaystyle \lambda _{i-\sigma +1}}$.

p. 164, equation (34), for ${\displaystyle (-1)^{i-s}{\frac {\left|{\underline {2s}}\right.}{2^{2}s\left|{\underline {i}}\right.\left|{\underline {s}}\right.}}}$ read ${\displaystyle (-1)^{i-\sigma }{\frac {\left|{\underline {2\sigma }}\right.}{2^{2}\sigma \left|{\underline {i}}\right.\left|{\underline {\sigma }}\right.}}}$

p. 179, equation (76), for ${\displaystyle i+l}$ read ${\displaystyle 2i+1}$.

p. 185, equation (24), for ${\displaystyle {\frac {x^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}$ read ${\displaystyle {\frac {x^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}-b^{2}}}=1}$.

p. 186, l. 5 from bottom, for 'The surface-density on the elliptic plate' read The surface-density on either side of the elliptic plate.

p. 186, equation (30), for ${\displaystyle 2\pi }$ read ${\displaystyle 4\pi }$.

p. 188, equation (38), for ${\displaystyle \pi ^{2}}$ read ${\displaystyle 2\pi ^{2}}$.

p. 196, l. 27, for ${\displaystyle e..e}$ read ${\displaystyle e_{1}..e_{2}}$.

p. 197, equation (10) should be ${\displaystyle M={\frac {E\,e}{f}}-{\frac {1}{2}}{\frac {e^{2}a^{3}}{f^{2}(f^{2}-a^{2})}}}$.

p. 204, l. 15 from bottom, dele either.

p. 215, l. 4, for ${\displaystyle {\sqrt {2k}}}$ read ${\displaystyle {\sqrt {2}}k}$.

p. 234, equation (13), for ${\displaystyle 2E}$ read ${\displaystyle {\frac {E}{2\pi }}}$.

p. 335, dele last 14 lines.

p. 336, l. 1, dele therefore.

p. 336, l. 2, for 'the potential at ${\displaystyle C}$ to exceed that at ${\displaystyle D}$ by ${\displaystyle P}$,' read a current, ${\displaystyle C}$, from ${\displaystyle X}$ to ${\displaystyle Y}$.

p. 336, l. 4, for '${\displaystyle C}$ to ${\displaystyle D}$ will cause the potential at ${\displaystyle A}$ to exceed that at ${\displaystyle B}$ by the same quantity ${\displaystyle P}$,' read ${\displaystyle X}$ to ${\displaystyle Y}$ will cause an equal current ${\displaystyle C}$ from ${\displaystyle A}$ to ${\displaystyle B}$.

p. 351, l. 3, for ${\displaystyle R_{1}^{2}u^{2}+R_{2}^{2}v^{2}+R_{3}^{2}w^{2}}$ read ${\displaystyle R_{1}u^{2}+R_{2}v^{2}+R_{3}w^{2}}$.

p. 351, l. 5, read ${\displaystyle +2\iiint (u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}})\,dx\,dy\,dz}$.

p. 355, last line, for ${\displaystyle S'}$ read ${\displaystyle S}$.

p. 356, equation (12), for ${\displaystyle {\overline {\left.{\frac {db}{d}}\right|}}^{2}}$ read ${\displaystyle {\overline {\left.{\frac {db}{dx}}\right|}}^{2}}$.

p. 365, in equations (12), (15), (16), for ${\displaystyle A}$ read ${\displaystyle Ar}$.

p. 366, equation (3), for ${\displaystyle {\frac {E_{2}}{r_{1}}}}$ read ${\displaystyle {\frac {E_{2}}{r_{2}}}}$.

p. 367, l. 5, for ${\displaystyle 2k_{1}S}$ read ${\displaystyle 2k_{2}S}$.

p. 368, equation (14), for ${\displaystyle J_{2}^{'}}$ read ${\displaystyle I_{2}^{'}}$.

p. 397, l. 1, for ${\displaystyle {\frac {D'}{E}}\delta '}$ read ${\displaystyle {\frac {D'}{E'}}\delta '}$.

p. 404, at the end of Art. 350 insert as follows:—

When ${\displaystyle \gamma }$, the resistance to be measured, ${\displaystyle a}$, the resistance of the battery, and ${\displaystyle \alpha }$, the resistance of the galvanometer, are given, the best values of the other resistances have been shewn by Mr. Oliver Heaviside (Phil. Mag., Feb. 1873) to be

${\displaystyle c={\sqrt {a\alpha }},\quad b={\sqrt {a\gamma {\frac {\alpha +\gamma }{a+\gamma }}}},\quad \beta ={\sqrt {\alpha \gamma {\frac {a+\gamma }{\alpha +\gamma }}}}}$.