OF ARTS AND SCIENCES. ' 319 



since {Aq) O and .'. {- /.q) <9 ; hence, in general, 



\qi = lXq. 



Hence 1 jTT = ^Ig, if (™ Z ?) O . 



Again Z.q~'^ = /.q, U V5' — ^ = — U V5' ; and hence 

 \q--^ = — \Tq — Z.q.\]Yq, 

 Yq — n—- — nVYq — n Z. ^'.UV^', if {ti /_q) O- 



We see then that, in general, n being integral or fractional, positive or 

 negative, and q being any quaternion, 



1^" = n\q, if — <(« Z ?) <+ a. (22) 



lO*^. The following results are deduced immediately from those 

 already obtained : — 



1(— q) = 1T(? — (9 — Z ?)UV^ 



= \q — auvg = 1^ + auv( — q\ (23) 



IKy = IT^ — Z ?.U V? = Kly, (24) 



\q-^ — — \Tq — A q.\^yq = — \q, (25) 



1| = 1T« - ITp^ + Z ;UV^«, (26) 



l«/3 = ITa -{- ITp^ + ( a — Z ^ ) U V«i3. (27) 



11°. Suppose a series of diplanar quaternions to be such that 

 Uy = U ^, U?' = UJ, LV = U^, etc., 



where a, ^, y, 8, . . . w, are any vectors whatever. Then 

 \Jd = \Jq"y, UcV - 1 = \Jq"j[3 - i = JJq"q', 

 J]q"q'q = U5p^ - i /3«- l = JJda - \ 



and, continuing this process, we shall finally have 



Jj{q^...q"q'q)=V-^, 



