170 
paper. In fact, I at first imagined that the proof of them was 
necessary to my purpose. They are obtained as follows :— 
_ “In virtue of the laws of combination of the imaginaries 
t, j, ky we have 
(ta +jb + ke)? = (— 1)" (a2 + B+ 0?) se, (8) 
Now, the coefficient of a?\b*c” in the left-hand member of this 
equation is 3 (2A, 2u, 2v), in conformity with the notation 
explained in p. 165: and the same coefficient in the right-hand 
member is plainly 
(A+pt+yv)! 
Alply! ° 
Consequently, we have the theorem I. 
(- pee 
(A+ m+ v)! 
Aluly! ° 
‘“¢ Multiplying both sides of the equation (8) by ta +jb+ ke, 
we get 
(ia +jb + he) ?e = (— 14 (ta + 7b + ke) (a? + B+ che, 
The coefficient of abc” in the left-hand member is 
= (2 +1, Qu, 2v); and the same coefficient in the right-hand 
member is 
= (2X, 2u, 2v) = (- LD i ga 
a ie aes 7 ae i 
We have, therefore, II., 
S(2X+1, Qu, 2v) = (- 1) See i, 
and similar expressions for j and /. 
« Again, multiplying (8) by (ta +jb + ke)? =— (a? + b+ c*) 
we get 
(ia + Jb: + fee) eee — (1) Mao (a? + 0? + ch) eet, 
The coefficient of a" 6°" c” in the left-hand member is 
=(2A+1, 24+1, 2v); whilst in the development of the 
right-hand member no such term appears, as all the exponents 
of a, b, c, must be even numbers. We have, therefore, III, 
