76 Proceedings of the Royal Irish Academy. 



r" " . dr = (5 + F) (r" . dA = ^±Sdq + - Vdq 



-+1 



n 

 m 

 and fiually 



n n 



dq + j^^^^ Vq-q""'^\.q^') V {Vdq : Vq); 



d.q"'=dr=r" \r" .drj^q'" \r" . dr 



n 11 n 



« 



(2.) Differentiate q"">. 



Let r = q >", then r'" = q~". 

 Differentiate, 



r'^-^.dr + r""-' .dr .r + ... + dr . r'""' = - q-" . dq . ?"' - q'"" . dq . q'''-. . .-q-'^ .dq . q-". 

 Multiply each member of equation 



— m 



r » = 5«+i 



by the corresponding member of the last equation and treat in a similar way 

 as in (1), the result is 



f^-? '"=-£?'" ^?+(-^?'" Vq-V.q '"jViVdq-.Vq). 



Therefore if a is any scalar either integer or fraction (or irrational, the 

 limit of fraction), positive or negative, 



d.q" = aq"-^ dq + (aq"-' Vq - V. q") V( Vdq : Vq). 



(3.) Differentiate log^ q. 



q"-! . 



loge q = Lim 



„-0 ?l 



therefore 



dq" 

 d loge q = Lim — = Lim 



n=(l n 11.0 



5"-' d^ + if-' Vq - —~) ^( v<ig •■ Vq) 



= q-'dq + (?-' Fq-V log, q) V( Vdq : Vq). 



(4.) Differentiate e«. 



1 



e' = Lim (1 + nq)" ; 



