EQUATIONS OF FROPAGATION OF ELECTRIC WAVES. 
i) 
taking (F, G, H) to satisfy the same system of clifiereiitial equations (8) and (!,)) as is 
satisfied ]jy (w, v, iv). The vectors (/, g, h) and (F, G, H) are connected ])y the 
equations 
(./> UJ^) = — G, H) 
— O' 
dt 
,{F, CT, H) 
( 11 ). 
It is also convenient sometimes to ([note the fundamental e([uatious in the forms 
o 
?f 
i_ a 
c- dt 
{/> vJ') = curl («, jS, y) 
(a, y) = curl (/, g, h) 
1 
(G). 
The (jiianfity (/i _q, /') "'cill sometimes be called the “electric disj)lacemenf ” ; it is 
the [)roducf hy 477 of the (|nantity so denominated by .Maxwelu ; tlie ([iiaidify 
(a, V, /r) w'ill sometijues he cidled tlie “ nuignetic dis[)hicement.” Maxwiu^iCs vector 
0 
[)otenticd would be expressed, in the above notation hy (F, G, FI). 
Special Types of Solulion, SoiD'ces ef Disturhauce. 
to. It has been [)ointed out that Kirchhoef’s method of iiitegTatioii of e([uation 
(I) de[)euds on the a})plication of a certain reci[)rocal theorem to two solutions of tlia.t 
e([uation, one of the two having the form (3). When we seek to apply a similar 
method to the system of e([nations (8) and (f), we are met at the outset hy the 
difficulty that no simultaneous solutions of the f)rm (3) exist, and hy the necessity of 
devising some forms of sohdf.ci, which shall become iidinite at the origin, and contain 
arbitrary functions. If we regard the f )rm (3) as corres[)onding to the solution r"’ 
of the e([nation V'<^ = the ap[)ro[)riateness of seeking for solutions of the system 
(8), Avhich correspond to splierical harmonics (jf order different fro)n zero, at once 
suggests itself. 
If in equation (I) we put 
v'here is a s[)herical surface harmonic of order n, and is a functi(.»n of r and t, u'e 
find for an equation which can be written 
0' i> {ri + 1) 
07“ ?•” 
('Ve.) = {r<h} 
By using the relations"^ 
* The ineUiod h;is l)eeii used hy R. R. Wkbb iu the discassicjii of RiecATi’s e(|UiiLiou; it ls iiidioiLed hy 
I. W. E. Cl.Al.sina:, ‘ Rhil. Tnms.,’ vol. 172 (1881), p. 801. 
VOL. CXCVll.—A. 
t 
