124 
ON CERTAIN RESULTS RELATING TO QUATERNIONS. 
[20 
(1 + \i + fij + vkY 1 (ix +jy + kz) (1 + \i + nj + vk) 
1 
i + x 2 + Y + v 2 
I ... (3) 
[ i [ x (1 + A 2 — fi 2 — v 2 ) + 2y (\/jl + v) + 2z (\v — p) ] 
; + j \2x - v) + y (1 — A 2 + Y - v 2 ) + 2z (pv + A) ] 
! + k [2x (\v + p) + 2y {pv — A) + z (1 — X J — p- + v 2 )] 
) 
= i {ax + a!y + a"z) 
+j (fa + $'y + /3"z) 
(4) 
+ k (yx + 7 'y + 7" z) ) 
suppose. The peculiarity of this formula is, that the coefficients a, /3,... are precisely 
such that a system of formulæ 
z, = <yx + 7 'y + 7 "z J 
denotes the transformation from one set of rectangular axes to another set, also 
rectangular. Nor is this all, the quantities X, p, v may be geometrically interpreted. 
Suppose the axes Ax, Ay, Az could be made to coincide with the axes Ax t , Ay t , Az, 
by means of a revolution through an angle 0 round an axis AP inclined to Ax, Ay, Az, 
at angles f, g, h then 
X = tan \d cos f p = tan cos g, v = tan cos h. 
In fact the formulæ are precisely those given for such a transformation by M. Olinde 
Rodrigues Liouville, t. v., “ Des lois géométriques qui régissent les déplacemens d’un 
système solide ” (or Camb. Math. Journal, t. iii. p. 224 [6]). It would be an inte 
resting question to account, à priori, for the appearance of these coefficients here. 
The ordinary definition of a determinant naturally leads to that of a quaternion 
determinant. We have, for instance, 
® , </>, x Wx"~ fix') + (fi'x- 4>x) + m " Wx' ~ fix)> 
/ # / / 
^ > <P > X 
// I // // 
» 9 > X 
&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),