414 Mr. Oliver Heaviside on the 
and, using the theorem (17), these give 
O=H+mVpVpuppu-'cH, 
0O=H+nVpVc"'pc- pH ; 
which, using the transformation-formula (18), become 
O=E + mp~'p(upcE) —u~'cH(p~"p )m, 
O= H+ ne7'p(c~!puH) —c—'wH (pe—'p)n; 
or, rearranging, after operating by mu and ¢ respectively, 
{(pu—'p)mc—p}H=mp(p~'pcE), 
{(pe~'p)nu—c} H =np(c~"pyH). 
Or 
_i p 
Sp Gepeoa 
H p ce 
cpu  (pe—p)u—n—e say.» . « (75) 
These give us the four simplest forms of equation to the 
wave. For, since ph=0=pH, we have 
py=90,  pye=O. 
Also, operating on (74) by wo'pe and on (75) by e—pp we 
get 
po pey.=1,. ec pey:=1, -. eee 
two other forms. 
71 and ry, differ from 8, and 8, merely in the change from 
o to p, and in the inversion of the operators. The two forms 
of wave (76) are analogous to (41), and the two forms (77) 
analogous to (42), inverting operators and putting p for o. 
Similarly, if the wave-surface equation be given and we 
require that of the index-surface, we must impose the same 
condition po=1 as before, and eliminate p. This will lead 
us to 
ocy,=0, opyjy=—m, ... . (78) 
corresponding to (60) and (65); and 
opy,=0, ocy,=—n,  . = . ene 
corresponding to (63) and (68); and the firsts of (78) and 
(79) are equivalent to 
ocH=0, opH=0; 
or the displacement and the induction are perpendicular to 
the normal. ‘This completes the first half of the process; the 
second part would be the repetition of the already given 
investigation of the index equation. 
