[ 353 ] 



DC X 



- z= ca^ ^ 1.2 nx no. hy. log. — _ i + — 



1.2 - na- a a 



X 



X " 



:: if M — no. hyp. loe. of which is i,y = Y M j- i. 2 - - 



I. 2. --« tf y r a ' -r 



a 



- nca X M — 1.2 n ca -{■ i. 2 nca x + 2- '^ ~ 



n — 2 2 B 



nea X ^ -|- cx' is the general equation of the fluents. 



As the above examples have confiderably extended the length 



of this trad, the fubjed fhall be concluded by a few obfer- 

 vations. 



The Theorem of Taylor may be more generally expreffed, 

 for if 5r be a quantity compofed of two or more independent 

 quantities x,jy, v, &c. then while x^y, v, &c. by flowing uniformly 



become x -\- x,jy j^j, 'z-' + v, &c. z will become z + -|- &c. 



There can be no diiflculty in applying what has been before done 

 to cafes of this kind. It may be worthy of remark, however, that 

 by this method when fluxions are fuch that the fluents are expreflTed 

 in integral powers, they may be found a priori: for if ;2: be a 

 fundion of x, j>, &c. where x, j, &c. are independent quantities, 



and Z the value of z when x, y, &c. = 0, then becaufe z='Z + z + 



z z . ' . 



h &c. and becaufe — , &c. are derived from z by taking 



1.2 1.2 JO 



Vol. VII. Y v the 



