Wang — The Differentiation of Quaternion Functions. 79 



= - Sf'g . Vcf" - Vf'q . Vq-'" + m Vfq . S{ Vq-'"-') + {m - 2) Vfq . V{ Vq-'"-') 



= - f'q . Vq-'" + \m-l + {-UVq^)'"*'} Vfq . Vq-'-' 



= - fq. Vq-'" +>l - 1 - (- 1)'» 1 Vfq . Vq'"'-', 

 if ?)!- is integer. 



(J J., (Sfq . Vq-") and (j „ ( Vfq . Vq''") are exactly the same as q {Sfq . Vq-") 

 and dqiVfq.Vq'^) respectively, except replace<^ q hyVdq,i.e.(jlSifqdq].Vq-'" 

 and aJV{fqdq).Vq-'" by a\S{fqVdq).Vq-'" and a \ '^^ if i'^'^l) -Vq'", or 

 - f'q . Vq-" and - 3 Sfq . Vq-" + Vfq . Vq-" by - Vfq . Vq-" and - 3 Sfq. Vq-'" 

 + 2 Vfq . Vq-'" respectively. The results are 



avgiSfq ■ Vq-") = - Vfq . Vq-'" + {m - 1 + (- l)'"i Sfq . Vq-"'-\ 

 and a vq{ Vfq ■ Vq-'") = - Sfq . Vq-" + {m-1 - {-l)'"]Vfq. F?-™"!, 



if m is integer. 



Now, we can successively operate by (j and (j ,, on fq without difficulty 

 by using the foregoing four formulae. 

 As fq = Sfq+Vfq; 



a,fg = - f'q -fq-^ vfq ■ Vq'' = - 2 s/v - 2 vfq -2Vfq. Vq-\ 



aifq = 2 f'q + 2 A + 4 Vfq . Fg- > + 2/V . F^' - 2 Vfq . Vq-' 



= 4 Sfq + 2 Sfq. Vq-' + 4 Vf'q + 6 Vfq . Vq-' - 2 Vfq . Vq-\ &C. 

 In general, 



af fq = (- 2)"-' (- 2^/(») q-{n- 1) ,S/("-i) q . Vq'' - 2 Vf"^q 



- (n + 1) F/(-')? . Vq-' + (w - 1) Vf"-')q . Vq-' | ; 

 because 



(Z"j(- 2)"-' ( - 2Sf"^ q-{n- 1] >S/{«-i) q . Vq-' - 2 F/^"' q 



- (?i + 1 ) F/C-') 9 . Vq-' + {n- 1) F/C-^'g . Vq-'] 



= (- 2)«-i {2/(''+')^ + (m-l)/("V • Vq-' + (n - 1) /S^/f^-O^r. Vq'' + 2f^"*')q + 4 F/<»)? . F?"' 

 + (71+ 1)/'") J . Vq-' - {n+ 1) F/(«-')^ . Vq-' - {n -1) /<""'>? . Fg'M 



= (- 2Y{-2Sf"^'^q-nSf"^q.Vr'-^ Ff""^q ~(n + 2) Vf^"^q. Vq-'+nVf"-'^q . Vq'']. 



Similarly, 



a r,fq = - Vfq -Sfq -2 Vfq . Vq-' = - Sfq - Vfq - 2 Vfq . Vq-', 



afqfq = Vf'q + Sf'q + 2 Vfq . Vq-' + 2Sfq . Vq' -2Vfq. Vq-' 



= Sf'q + 2Sfq . Vq-' + Vf'q + 2 Vfq . Vq-' - 2Vfq . Vq-', &C. 

 In general, 



a"r,fq = ( - ir{'?^<'% + ('1 + ^~'^]^^^ )s/(»-')q . Vq-' + Vfi")q 



+ in + ^— ^^') Vf"-')q . Vq-' -(n+ ^~^^^"~^ ') F/("-=)? . Vq-'\ ; 



