= Olam 
377 
tion the forms of simple quaternions, and their substitution in 
the series reduces it to a quaternion also, the coefficients 
W, X, Y, Z being expressed by infinite series. 
‘* From the form of f(Q), as expressed in (1), it is evi- 
dent that we shall have Qf(Q)=/(Q) Q. Now substi- 
tuting for Q and f(Q) their values, viz. 
Q =wite t+jy +hz (2) 
JS(Q) = W+iX4+jY+kZ 
we have | 
(wt ia+jy+kz) (W+iX+ jY¥+ kZ) 
=(W+iX+jY+kZ) (wtia+jyt+kz) (3) 
an equation which, it is to be observed, would not be true 
unless the quaternions w + tx + jy + kz and W+iX +jY+ kz 
were functionally related. Multiplying out, and attending to 
the rules of quaternions, we have from (3) 
w—aX-yVY-2zZ7+i(wX + We+yZ- Yz)+j(wY+ Wy 
+2X—- Zn) +k(wZ+ Wz+uY- Xy)=w?- Xax- Vy 
— Zz+i(Wa+ Xw+ Yz-yZ)+7j(Wy + wY+ Za - 2X) 
+k(Wz+wZ+ Xy-2Y). 
Equating coefficients, we find 
yZ-Yz=0, zX-Ze=0, tY- Xy=0, 
or Ki oR 4 V. 
ae ee suppose. 
We are therefore permitted to assume 
S(w + ix + jy + kz)=W+ Vein + jy + kz) (4) 
and it remains to determine W and V. 
“* Now f(Q) being by hypothesis of the form 24, Q", let 
us seek the special forms of W and V for the particular case 
in which f(Q) = Q". 
‘¢ Tn virtue of (4) we may then write 
Q" = Wr + Vn (ix + jy + h2), (5) 
Q™ = Wis + Van (te + jy + he). (6) 
2N 2 
whence 
