40. Mr. A. Cayley on Algebraical Couples. 



X2' 



i» = y» + Sj f^^' 



X + X Y = ± — (y ± >^ 8) (.r ± x^) (^'i + xyi)ii '."in;>n • 



X 



D'. ,2==_g,_^ 



x^-^ 



- i 1 ' , 



;^= X^6» + y; , , 



. • X±xY= + — (y±xg.)C^-=F ^^)(-^'i^%'!R'^^'' ^T' 

 ,oiJ ^ •■ - ■' oi Jr>ir// 



There is a system more general than, (A,) having- /a ising/ifi 



linear modulus <7 (5X4- Y): this is rwojlij lyod ne gnininlas 



E. ,2 = « (. _ 5;) + ^nf^^ f II" :}fi 9Wi«'v 9low 



(,.^ y — « I' y^ + V;^,;fii,i^o jibyob ihjmji > 



,ii' J, = y(i-5j) + flyj,, ..rn.n' 'I liinij ^■:,mh 



' .lJ6ov< adi bin. f = r' (* — ^iO + !77 . fiosrrcwl .a.K 



. ■ ,,.,,, . f X + Y = <7 (6 jr + _?/) (5.Av-K34j)4^.-i.':; -<iooa ^minv/ 



or, without real loss of irenerality, ^/ {' "'* ^^^^ -3^^"^^^ ^^f f"* 

 ■T', ° 2_ ''•• 3 it Ivj'fj'/oa 'jjjonf nu fit hnj; 



^* ' -«^,{<rir vr:f, odi lo J29T 9dj 



■ . ■■,/■.: y = «'» Of;-ol .ffi.j? ^C 



J2 = y' , + qj. 



To complete the theory of this system, one may ad^^'uie 

 identical equation ^ 



1 V y(62_Ma)/ a'-fiy \/ a'-M T' 



x + 



.•here M = Hy-H) - Jc^-^c^) ^ 



K y — « y 



\ a! fl V 



By determininij the constants, so that -i s> = — — rAi 



a'^&y' -^ 



= - _ g ? the system would reduce itself to the form A. .j,jH 



Cam bridge, April 23, 1 845. 7 '^^ ^ i 



• . hllJ; 4otl 



