921] 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
203 
or (what is the same thing) an indefinite number of arbitrary constants. In fact, 
writing for shortness z=x + iy, z x = æ x + iy 1 , and z x for the conjugate functions 
x - iy, x x - iy x ; also <£ (z)_ for a function of z involving in general imaginary coeffi 
cients a + ib, &c., and c/> (z) for the like function with the conjugate coefficients 
a — ib, &c. ; then if we assume 
cf>(z) 
where m is any positive or negative integer, this implies 
. = »(«)_ . 
U 
consequently, if x 2 + y 2 — 1 = 0, that is, zz = 1, or z = ~, we have 
z 
à' 1 
Z m (f) 
g)"*w + (z) ’ 
or z 1 z 1 = 1, that is, x x 2 + y x 2 — 1 = 0. 
In a slightly different form, taking a, /3, &c., to denote any imaginary quantities, 
and a, /3, ... the conjugate quantities; assuming 
0 (z) = (z - a) (z — /3) ..., 
and taking m for the number of factors, we have 
z 
1 (1 — az) (1 — fiz) ... ’ 
and then (repeating the demonstration) we have 
(z—a) (z-/3) ... 
(1 — az) (1 — (3z) ... ’ 
which, writing therein z = -, becomes 
1 
a 
z 
z "‘ _ (1 — a^)(l — /3^)... _ 1 
1 - 
1 - 
/3 
(z — a.) (z — ¡3) ... z 
and consequently, if z =-, then also z x = — as before. 
We may in the expression for z x introduce a factor -=, or, what is the same 
A 
thing, a factor A which is such that AA = 1. In particular, we thus have the solution