74 Proceedings of the Royal Irish Academy. 



But, similar to dq, dfq can only be resolved into one set of components, a 



quaternion IHfq, i.e. 



dfq+Vfq.V{Vdfq:Vfq), 

 and a vector J_ Axfq, 



-Vfq.V{Vdfq:Vfq). 



Therefore we can equate the eoplanar quaternion part 



fq . [dq + Vq . V{ Vdq : Vq)] = dfq + Vfq . V{ Vdfq : Vfq) 



= dfq + Vfq . V{ Vdq : Vq) 



('.• Accfq = + Axq, dAirfq = ± dAoeq, 

 and V(Vdfq : Vfq) = dAafg : A:rfq = dAxq : A.rq = F( Vdq : Vq) ). 



This gives dfq =f'q . dq + {fq . Vq - Vfq) V(Vdq: Vq). 



In addition to the foregoing proof, the differentials of some particular 



functions of q can be found in other ways. The results are the same as 



those deduced from the above equation ; therefore they can be used as the 



verifications of it. 



» 



(1.) Differentiate q'", where vi and n are positive integral scalars. 



Let r = q>», then r'" = q". Differentiate, 



r"'-' . dr + r'"-\ dr.r + ...+ dr. r'""' = q"-' . dq + q"'"' .dq . q + . . . + dq . f'K 

 Multiply each member of equation 



m 



— m -f — 



r " = q-'"-' 

 by the corresponding member of last equation, 



m 1H in 

 1 ---2 m 



r" .dr + r" .dr.r + ...+ r" .dr.r"'-' 



= dq + q'^ . dq . q + . . .+ q~"*' . dq . q"''^. 

 Denote Q + r"' Qr + r'- Qr + . . . + ?•-"•" §?'"'"' by F, Q 



and Q + q-' Qq + q-' Qq' + . . . + <?"'»' Qq"-' by F^ Q, 



then Fi-"~\dr) = ^''^^- 



Hence every term of F, and F^ is a conical rotation of the operands 



therefore 



SF, = F,S=mS, VF,=F,V, 



and SF, = F,S = nS, VF, = F,V; 



thus mSyr" . dr) = nSdq, 



and F, V\r" . dr) = F, Vdq. 



