EQUATIONS OF PEOPAGATION OF ELECTPJC WAVES. 
4v/o = If ^3 |(((7 - mf) + ~{lg - m/)| - \{7if-lh) + ~(n/- 
•2 3(7/2+ .3)1 ^ X . . -nI 
~3 -- nv) + ~{mw - 
17 
III) 
+ 
o , o 
ir + 2 " / .. ..\ 
- (mw — nv) 
,,3q2 V / 
+ ^ j 3 {nu — Iw) + 3 ^-(mi — hb) + '*^{nu — lib) 
o' ' ■ c-^ 
r V . 7-3 
O'Z r T 1 
+ ,10 ~ *'*“) + 3 / y-v — ™’‘) + yv — m-ii) ) 
(38), 
in which the values of v, u, ... A, at any point of tlie surface, at time t = — o/c’ 
are to be calculated, and the integral formed with these values. 
16. In this ecjuation, the point 0 is the origin, and the point [x, y, z) is on the 
surface We can express the value of f, at any jooint [x, ?/, z), and at any time A by 
suitable changes. We have to write x — x, y' — ?/, z' — z for x, y, z, and, in the 
expressions for u, u ... in terms of x, y', z! and (, we have to substitute for A 
t — r/c, where r is the distance between the points [x, y, z) and {x\ y', z'). Further, 
when the form of f has been obtained, the forms for g and h can be written down liy 
symmetry, and the forms for u, v, iv can be deduced from those for f, g, h, liy writing 
F, G, H instead of u, v, tv, and u, v, iv instead of f g, h. 
It is convenient to have, for reference, the explicit expression of the results. For 
u, V, IV we have 
47r?( = dS 
'If ~~ i/ r T zz^ r T 
- 3 ^|{//; — riiu) + - [Iv — mu)\-\- |(7m — hv) + -^{nu — Iw) 
(// - 7/')' + (2 - 
+ 1 u; — 3 
^'\2 
(mH — nG) H —[niYi — tiQ) 
+ 
iy - y'f ^ - z'f , ^ 
-„ „-(tuH — wG) 
7”V” 
G - -F) [y - y') 
(x - o:')(z - z') 
S{nF - /H) + 3 ’■ (7rF - /H) + {nf - /H) 
C C" 
3 ((G - 7uF) + 3 -((G - mF) + - wF) 
4 Try 
= IfdS 
. ^ 
{mtv — nv) -\— {niw — nv) I + 
x — x 
+ 
+ 
(x - x') {y - y') 
\)-s 
7 
{Iv — nm) + - (/y — mu) 
3 (mH — nG) + 3 - (mH — nG) + ^ (mH — /iG) 
{z - z'f + {X - O-'f 
|(„f-«h) +A«f- m) 
(.V - y') (2 -«') 
+ ----- I 3 (/G - mF) + 3 ((G - mF) + (/G - mF) 1 
7 (_ C C“ j _ 
VOL. CXCVIL—A. 
D 
