p 

 A. 



1818.] Lacroix's Differential and Integral Calculus. 205 



marked E and F ; and as by their application many fluents may 

 be found with very little trouble, your printing them with the 

 accompanying illustrations in the Annals of Philosophy will 

 much oblige Your most obedient servant, 



James Adams. 



Formula. 

 C V -1 dx (a + b x n f — 



i»- « (a + Jx" ) p +' — a (m—n) Jx' n - n - 1 d x (a + b x ) 

 b(P n + m) 



/V- 1 dx (a + b x n Y = 



x m (a + 6 x n ) p + a np f x m ~ 1 d x (a + b x n ) p — * 



5 1 — ; B. 



F n + m 



x m ' 1 dx (a + b x n f = 



x'"(a + b x n ) p + l — b (m + n + p n) Jx m + n ~ l dx (a + b x n ) p 



■ ' • • > • ,v/. 



a in 



.r" 1 -" ((? + Ji") p + ' - ftn(P -f 1) fx m ~ 1 dx (a + b x n ) v 



— ■ E 



m — n 



f 

 f 



p x m ~ l dx (a + 6a-'') p - , = 



x'" - ' (a + b x" ) p (m — ti) fx m ~ n ~ ' d x (a + b x n ) p 



7 F. 



n b p 



Illustration. 



The following examples are taken from " Dealtry's Fluxions/' 

 chap. xxii. By a little attention it may easily be seen that the 

 Jirst thirty, thirty-four, and thirty-siith fluents may be deter- 

 mined by formula A ; the thirty-first and thirty-fifth by formula 

 C ; the thirty-second and thirty-seventh by formula E ; the 

 thirty-third by formula F ; and the thirty-eighth by formula A 

 and B. Some of the fluents in the succeeding section of the 

 same chapter, and many others, may be determined by the 

 preceding formula. 



Fluent 4. — To find the fluent of -, where m is a whole 



x — a 



positive number. 



/x m dx /-• 



— =J i'Jx(.t-fl)-' 



x m a I xm— ' d x 



= — . -f. _: » 



m x — a 



