410 Mr. Oliver Heaviside on the 
vector of the wave-surface, 
PPS=P > sein os een || 
But o the vector of the index-surface being = Nov-!=pv-?, wii 
have by (47), dividing it by v’, 
opal... .:. \si..,, 1 arr 
To find the wave-surface, we must therefore let o be vari. 
able and eliminate it between (48) and any one of the index: 
equations. This is not so easy as it may appear. 
General considerations may lead us to the conclusion thati 
the equation to the wave-surface and that to the index- cue 9 
may be turned one into the other by the simple process oii) 
inverting the operators, turning c into ¢~! and p into p—". 
Although this will be verified later, any form of index equation) 
giving a corresponding form of wave by inversion of operators, 
yet it must be admitted that this requires proof. That it is: 
true when one of the operators c or w is a constant does noti 
prove that it is also true when we have the inverse compound’ 
operator {(oco)u-!—nce—!}—! containing both ¢ and p, neither: 
being constant. I have not found an easy proof. ‘This will 
not be wondered at when the similar investigations of the 
Fresnel surface are referred to. Professor Tait, in his ‘ Qua-. 
ternions,’ gives two methods of finding the wave-surface; one 
from the velocity equation, the other from the index equation. 
The latter is rather the easier, but cannot be said to be very 
obvious, nor does either of them admit of much simplification. . 
The difficulty is of course considerably multiplied when we: 
have the two operators to reckon with. I believe the following - 
transition from index to wave cannot be made more direct, or | 
shorter, except of course by omission of steps, which is not a 
real shortenin g. 
Given ch A . 
aoe (cuc)e!—mp-” (49) =(39) bos, 
o8i\=0, . . . « .. OO=@a—— 
po=l, .... .. . O1l= 4a 
Eliminate o and get an equation in p. We have also 
poe Py=1,. . ... O24 
which will assist later. 
By (49) we have 
o=(cpo)c'Bi—mpB,, . . . «en 
Multiply by 8, and use (50); then 
O=(ouc)(Bic-'B,)—m(By-18,). . . «  (4) 
