1915-16.] On Systems of Partial Differential Equations, Etc. 307 
and noticing that 
X cos 9 + Y sin 0 + i7* = ic[(^ 2 - V1V2) cos 0 + (^2 + £2^1 ) s i n 0 ] 
+ .^[(^2 “ V1V2) sin ^ “ (^2 + C0S #] “ iz (€l%2 + W2) + ~ 
where 
$ p = aj_p cos 0 + 2/p sin 0 + i’z , 77^ = sin 6 — y p cos 6 + ct p (p = 1, 2), 
we see that Y can be expressed in the form 
/°27T 
v = J o 0)<W. 
It follows then that Y is a right-handed double wave-function * of the 
variables aq, 2/1, ^ ; x 2 ,y 2 ,z 2 ,t 2 . 
Again, if we use C (0) to denote the vector with components (cos 0, 
sin 0, i) it is easy to see that 
grad V = l ^ ; Vv 7/2 ; °W e ) de= M sa F> 
and that this complex vector M satisfies the two sets of partial differential 
equations 
rot, M = -+ + div, M = 0, 
c ot 2 
rot„ M = - - — , div„M=0, 
2 e dt 2 ’ 2 
where the suffixes indicate the set of variables with respect to which the 
differentiations are made. Writing M = H+iE where E and H are rea 1 
vectors, we find that these vectors satisfy Maxwell’s equations in each 
the two sets of four variables x ,y p ,z p ,t p (p = l, 2). 
The case in which 
X = c{x x t 2 - x 2 t ± ) + %,z 2 - y 2 zf), 
Y = c(y 1 t 2 — 2/2^1) + i{ z i x 2, ~~ z 2 x i )> 
z = c(zf 2 — zf'x) "t i(xgj 2 ~ *^2^1)’ 
R 2 - (x x x 2 + y x y 2 + z x z 2 - c\t 2 ) 2 - (x\ + ||+ z\ - + yl + z\- cHl) 
is of special interest. If in particular we write V = g, so that the com- 
ponents of E and H are given by expressions of type 
K 6 
we obtain a specification of a vector field with some interesting properties. 
The quantity R 2 vanishes and M becomes infinite when the equation 
K + Yr 2 ) 2 + + hy 2 f + (z 1 + A z 2 f = c 2 (q + Xi 2 f 
gives two equal values of A. This happens when the common tangent 
* For a definition of this function see a paper by the author, Bulletin of the American 
Mathematical Society , April (1916). 
