[ 



Mr, A. Cayley on Algebraical Couples^ 

 r 



50 



C. ,.=:__(U^ + y(X + ^)).-^i 





/ = (A + /x, y + Aft 8) * — 77 



r 



J 



X + ft Y = — (7 + /A S) (a^ + ^ j/) (a^i + Aj/i). 



D. 



''=-s«' + 3^(rH-SA + ^)j 



/= -.Xii.li + {y-VX + fs.l)j 



X + XY = — (7 + AS) (^ + /tj/) (^, + A3/1) 



X + ,A Y = -i (7 + /* 8) (^ + A ?/) (o-i + /^ j/i ). 



The formulae are much simpler and not essentially less ge- 

 neral, if jct = — A. They thus become 



A'. 



H. 



''^ = 8, + -Ji 



/=^'s, + 7i 



X± A Y = ± y(7±aS) . {x±\y) (:ri±Aj/j). 

 (Two linear moduli.) 



*i=i» = 7< + 8j 

 x±ay= + Y(y±^^)(-*' + ^^)('^i + ^i/i)- 



