I 334 ] 



a. The number of fluxional fadors to be affixed to the r<h term 

 is r, and the fum of their exponents is to be «. Thus if 



z 



xf X A-? X &c. be one of thefe produds /> + 2 f + &c. = «, and 

 f 'T q ■\- &c. = r. 



3. The coefficient of x^ ^ A•^ &c. is as ftated in Prob. I. 



Example. The fourth fluxion of the fine of 3.V, when x = 

 is 3 X — 3. 2. x^ X. 



CoR. The n"' fluxion of the cofine of w^ is had by fubftituting 

 in the above feries for the cofine of mx, — j, of ?« x, and for s, 

 mx, cs, mx. 



Tlie Application of the preceding Problems. 



Problem 5. The relation of two quantities being exprefled 

 by one or more equations to find the value of one of them in a 

 feries afcending by integral powers of the other. 



Solution. Let x and y be the two quantities to find x in a 

 feries, afcending by integral powers ofj' Compute from the given 

 equations, by help of the preceding problems, the values of x (A), 



* (B), X (C), &c. when J* = a given value as a andj' =7 — a makings 



flow 



