IX. On Algebraical Couples. By Arthur Cayley, Esq.^ 

 B.A., F.C.P.S., Fellow of Trinity College^ Cambridge'^, 



TT is worth while, in connection with the theory of quater- 

 A nions and the researches of Mr. Graves (Phih Mag., 

 No. 173), to investigate the properties of a couple tx+jy, in 

 which ijj are symbols such that ^ 



ij = «! » + ^V" 



If I X +jy 1 0^1 +7>i = i X +7 Y, then ,^ , -^ ,^ 

 X = ad^j^i + «' ^j/i + r^ii/ + y* j/yi,"^ 



Imagine the constants «, ^... so determined that ix +J'y 

 may have a modulus of the form K (a; + \y) {x + [uy) ; there 

 results one of the four following essentially independent sy- 

 stems:— 



*i =i * = y ' + V ' '' * "^ ~ '.^' 



/ = _ ;,^g/ + (y + ;, +^S)y • ^ "'v. 



X + \ Y = -i (y + X 8) (.r + Xy) (^1 + A^ j 



-< 1 



X + /^Y =— (y + /x 8) ix + iM,?/) (^ + luy,). 



The couple may be said to have the two linear moduht. ;, „ ^.. 



_L(y + x8)(.c+Ay):^(y + ^8.)(^ + |t.^); 



as well as the quadratic one, 

 1 



y + khy + [j^hx + \yx + fjt,yi 



A Ur 



the product of these, which is the modulus, and the only mo- 

 dulus in the remaining systems. 



B. ,2^_8, + -L(y + 8A+7)y d 



A JIC 



*i=i' = y« + ^i 



* Communicated by the Author, 



