Sir William Rowan Hamilton on Fluctuating Functions. 299 



Dj* tan being understood to denote the value which the i'" differential coefficient 

 of the tangent of a, taken with respect to a, acquires when o := ; thus, 



1 = Dj tan = 3Dj tan — d/ tan ] 

 = 5DotanO — 10D„HanO + D„*tanO. J 



Finally, if we make y^ = cos a, and attend to the expression (p''), we obtain, for 

 the secant of an arc /, the known expression : 



7_v - 2(-l)"+' 

 sec I - 2.(„,_„ ^2^ _ !■) ^ _ 2/ '■> 



(f) 



which shows that, with the notation (niQ, 



iecl=.<i)J°-\- (i)^P-{- wj^ -{- ...y (u'') 



and therefore, by the relations of the form (n^). 



/ - 1 (1 - (^- 1 D„)^*secO) = (1 + /- 1 D„)=*tanO ; (v^ 



thus 



1 = secO = 2D„tanO — Do^secO 1 



r y" } 



= 4d„ tan — 4d/ tan + Do* sec 0. J 



Though several of the results above deduced are known, the writer does not 

 remember to have elsewhere seen the symbolic equations (r*'), (v''), as expressions 

 for the laws of the coefficients of the developments of the tangent and secant, 

 according to ascending powers of the arc. 



[20.] In the last article, the symbol r was such, that 



and in article [11.], we had 



1 + 2(„)'; Q„,» = N2„„+„ N,-'. (y 



Assume, now, more generally, 



V^s^^ = N^„Nr*; (zO 



and let the operation v^ admit of being effected after, Instead of before, the 

 integration relatively to a ; the expression (m") will then acquire this very gene- 

 ral form : 



2q2 



