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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. 
‘«s 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 =wtta +jy +hz . (2) 
S(Q) = WH iXsjY+kZ 
we have ; 
(w+ ix+jyt hz) (W+ix+ jY¥+kZ) 
=(W+iX+jV+kZ) (w+ixn+jyt+hkz) (3) 
an equation which, it is to be observed, would not be true 
unless the quaternions w + ix + 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—-yY-2Z7+i(wX + We+yZ- Yz)+j(wY¥+ Wy 
+2XK—Zx)+kh(wZ+ WeinY- Xy)=w — Xx- Vy 
—Zz+i(Wa+ Xw+ Yz-yZ)+7j(Wy + wVY+ Zu - 2X) 
+k(Wz+wZ+ Xy-2Y). 
Equating coefficients, we find 
yZ-Yz=0, z2X-Z«=0, «Y- Xy=0, 
ie Xi le 4 se 
eG , suppose. 
Weare therefore permitted to assume 
S (w+ iv + jy + kz)=W+ Via + jy + kz) (4) 
and it remains to determine W and V. 
«‘ Now f(Q) being by hypothesis of the form =4,Q", let 
us seek the special forms of Wand V for the particular case 
in which f(Q) = Q”. 
‘‘ In virtue of (4) we may then write 
Qr = W,+ Vn (ia + jy + hz), (5) 
Q™ = Was + Via (ax + Y + kz). (6) 
2n 2 
whence 
