1895-96.] Prof. Tait on Electro-magnetic Wave-Surface. 165 
On the Electro-magnetic Wave-Surface. By Prof. Tait. 
(Read April 2, 1894.) 
We may write the electro-magnetic equations of Clerk-Maxwell 
as 
4>0 1 — YV0 2 , if/0 2 = — V V 0 l 
For plane waves, running with normal velocity va— — /x _1 , we 
have 
0 1 = e f(vt + Sap) , 0 2 — y] f(vt + Sap) 
whence at once 
<$>e = V fjLrj , if/rj = — V/xe , 
so that S/x<^>e = 0 , S fjof/rj = 0 . 
[For the moment, we assume that cf> and i jr are self-conjugate, so 
that a linear function of them is also self-conjugate. And we 
employ the method sketched in Tait’s Quaternions, §§ 438-9.] 
We have nxf/ -1 ^ = Yif/fiiJ/r] = — "V.j^/xV/xe 
or ^/xSei/qx = 7icfi€ + S fiij/fi . if/e — we say. 
Thus we have, to determine /x, the single scalar equation 
S . /x<£oi7 _1 i/qx = S/x(?^ _1 + S/xi^/x . ^~ 1 )~ 1 /x = 0 . . . ( a ) 
This is the index-surface, and the form of uj shows that it has 
two sheets \ there are two values of Tp, for each value of U/x. 
The tangent plane to the wave is S/xp = - 1 . . . (b) 
To shorten our work, introduce in place of e the auxiliary vector 
r = to _1 i/px = e/Sei/px , 
so that if//ji = UfjiT + S fjoJ/fx \[/t .... (c) 
(a) may now he written 
S/x^t = 0 . . . (a) 
