Mr. Cay ley on certain results relating to Quaterniojis. 1 43 



an axis A P inclined to A a:, A j/, A z, at angles j^ g, h, then 



X = tan ^ Q cosj", ju, = tan ^5coso-, v = tan|9cos^. 

 In fact the formulae are precisely those given for such a trans- 

 formation by M. Olinde Rodrigues Liouville, t. v., "Des lois 

 geometriques qui regissent les deplacemens d'un systeme 

 solide" (or Camb. Math. Journal, t. iii. p. 224). It would 

 be an interesting question to account, u priori, for the appear- 

 ance of these coefficients here. 



The ordinary definition of a determinant naturally leads to 

 that of a quaternion determinant. We have, for instance, 



=ot4>' — '^' <Pi (6.) 



^', ^', x' 



^,<J>' 



=^{^'x"-<p"x') + '^{<P"x-<Px") + '^'M-<P'x)y 



&c., the same as for common determinants, only here the 

 order of the factors on each term of the second side of the 

 equation is essential, and not, as in the other case, arbitrary. 

 Thus, for instance, 



(7.) 



but 



'.Tjs ts' — zsr ot' 



/_ 



0, 



j' — •ct'-btstO. 



(8.) 



Or a quaternion determinant does not vanish when two verti- 

 cal rows become identical. One is immediately led to inquire 

 what the value of such determinants is. Suppose 



■u!-=x-\-iy-\-jz-\-k'W, isy' = a,'' ■}• i 7/' +J z' + k ijcf , &c., 

 is it easy to prove 



•or ttT 



= —2 





\ 



W, ■ST, "57 



= -2 



?5 h 



"BJ 'OT ZT 'ST 



-tsj' -ct' ot' -ct' 

 tJ7" w" ^" «;" 



k 



w 



^', y, z'', tt/' 



^•'j y, 



>. 



(9.) 



(10.) 



(U.) 



'BT •GT w HJ 



Or a quaternion determinant vanishes when four or more of 

 its vertical rows become identical. 

 Again, it is immediately seen that 



:;ri}.(-) 



•nr, <$) 



+ 







