Macfarlane — Differentiation in the Quaternion Analysis. 209 



The problem is reduced to finding the value of v^' and of vp- 

 When p is constant, V-S = PV^- According to Hamilton and Tait 

 Vr = p ; hence, for the case of p being constant v^ = p"^, which has 



the reduced value of - 1 . Eut if we take v^ = -> in the case of p 



P. 



being constant, V-^ "^iH t»e p/p = 1 absolutely, that is, independent of 



the principle that p^ = - 1 , and of the condition that p involves the 

 y/ - 1 . As xi+ yj + %h is the sum of three such vectors with constant 

 axes, _ [^(xi)\ d(yj)\ d(%h) \\ 



Consider next the general case when both r and p are variable. 

 We have V (^p) = 1 + ryp. 



As one term of this equation is an absolute number, I infer that the 

 other two terms are absolute numbers. There is reason to believe 



2 . . 2 



that V (rp) = 3 ; which makes vp = -• According to Tait, VP = — > 



r r 



and v(^p) = - 3. 



Let p - = 1 be differentiated in situ, that is, as an ordinary product 



P 



of vectors ; ,1 , / 1 \ ^ 



dp- + pd[- ] = 0, 



P \PJ 



therefore di -] = — dp - - — dp, 



\PJ P P P 



because p and dp are at right angles. 



[\\ 1 



Hence d\ -\l dp = — • 



\PJ P" 



In what follows, I propose to apply the above principles, viz. that 



2 



r 



d\ -\ I dp = —, W = -■> and vp = 

 \P) P P 



First apply them to find y- - which is known to be 0, 



1 1 



V - = - - v^ = 

 r r- 





[~pI~ r^P 



,1 1 

 r r^p 



2/r 

 Ap, 



Ip r^Ap^jr 



2 2 



y3p2 ^3p2 



