(1.) 



142 Mr. Cayley on certain results relating to Qiiaternions. 



Journal, as one that might refer to the same subject. It may 

 perhaps be as well to notice that the investigations there con- 

 tained have no reference whatever to Sir William Hamilton's 

 very beautiful theory; a more correct title for them would 

 have been, a Generalization of the Analysis which occurs in 

 ordinary Analytical Geometry. 



I will, with your permission, take this opportunity of com- 

 municating one or two results relating to quaternions ; the first 

 of them does appear to me rather a curious one. 



Observing that 



{A+Bi+Cj+Dk)-' 



= (A-B^-C;-D/?:)-(A2 + B2 + C2 + D2) 



it is easy to form the equation 



( A + B / + Cj+B k)-' {u + Si + yj-\- U) (A + Bi+Cj + D k) ^ 



1 



""A^+B^ + C^ + D^ 



f «.(A2 + B2 + C2 + D^) 



J +/[g(A^+B^-C^-D^) + 2y.BC + AD + 2a. BD-AC] 

 ^ +j[2g(BC-AD) + y(A2-BHC2-D2) + 2S.CD + AB] 



^ + /t[2§(BD + AC) + 2y(CD-AB)+8.A2-B2-C2+D^]. 



which I have given with these letters for the sake of reference ; 

 it will be convenient to change the notation and write 



( 1 + \i + [jj+vk)-'^ . {ix +jy + kz) ( 1 + y.i + ijuj + v k) 1 

 _ 1 



r ? [j:(l+A2-p,2_v2^ 4. 27/(X^ + v) + 2z{Kv-il)'] + 



U [2 ^ ( A V + ja ) + 2 j/ . ( JU, V - X) + ^ ( 1 - A2 ~ 1^2 _,. y 2 )-] 



i (a X + a! y + a" %) "] 

 +j,{^x + ^'y + &<z) \ (4.) 



. + k{yx+ 7' j/ + / z) J 

 suppose. The peculiarity of this formula is, that the coeffi- 

 cients «, /3 ... are precisely such that a system of formuhae 



x^=^ ax + ai y + a";^:! 



3/,= & X ■\- g'j/+ g"-^ r- (5.) 



z^ — yx-\- y'y + y" ;s J 

 denote the transformation from one set of rectangular axes to 

 another set, also rectangular. Nor is this all, the quantities 

 X, («., V may be geometrically interpreted. Suppose the axes 

 A.X, Ay, A^ could be made to coincide with the axes A.r^, 

 Ayp Ax:, by means of a revolution through an angle Q round 



•, (2.) 



h (3.) 



