258 PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIYE GEOMETEY. 
49. A function which w the inverse of its conjugate is in general reducible in an 
infinite variety oj ways to the ‘pi'oduct of a self-conjugate function and a rotator. 
Because gg' — 1 in the notation of Art. 47, the conditions (198) that g should he 
reducible are 
^(l) = G(l), where G'= 1, G = G'.(199), 
for simplicity writing 
l+.9(*) = «’ 1-P'(1) = 6. . . . . . . (200); 
it is evident from the last equation that 
Ga = a, Gh = ~ b, Scdj = 0.(-01) I 
so a and b are united points of G, and conjugate with respect to the unit sphere. 
Take any point c in the polar plane of b, and any point d in the polar line of ac ; 
and assume 
Gc = c, Gd = — d .(202); 
then the function determined by the four relations (201) and (202) is self-conjugate, 
and its symbolic equation is G' — 1 = 0. By the construction it follows that 
S«6 = S6c = Sad = Sch = 0 .(203), 
and the function is consequently self-conjugate. 
We have now determined a self-conjugate function, one of an infinite number, 
which satisfies (199), and the 23roposition is jjroved. 
The rotator corresponding to G is of course 
lX = G~^g = Gg .(204). 
50. The results of recent articles establish the possibility of reducing an arbitrary 
function to the form 
f=FGU . (205); 
where F, G, and R satisfy the equations 
T^=ffi, F-fi{l) = G{l), G2=l, R = GF-V'. . . (206); 
and by analogous 23i'ocesses the function may also he reduced to other forms such as 
G^F^Rj, but on these we need not delay. 
51. An arbitrary function may be reduced to a quotient or i^roduct of two self¬ 
conjugate functions. 
Assuming 
/= Fr^Fi.(207), 
it a23pears that the united points of (com 2 )are Art. 30) satisfy the equations 
Fj« = ^jFoU ; F-f = t jFd)-, FjC = ^gF^c; Fjh = ^4Foc/ . . (208); 
