[ 337 ] 



quantity, the exponent of which is jr m is to be added. The 

 fecond term is the fum of all the quantities 3a^/2, 3. 2y^e, fo that 

 the fum of the exponents of each quantity — m : and when ?n is 

 a multiple of 3, the cube of the term, the exponent of which is 



— is to be added. 

 2 



Now when ?' = 0, x^ -\-gx = o, and the values of x are 0, 

 + ^ — ^, fubftituting thefe values In the values of a, b, c, &c. found 

 above, and r for r, we have the three values of ^ + ^ + <: + &c. 



X 



— X + — "^ &c. the three increments of x, while r from be- 



2 



comes r. Let thefe values be A, B, C, and the valu es of x arc 



A, v/^ + B, — ^/':^ ^ C. 



The preceding is given as an example of the method, and not to 

 fhew its fuperiority to others. Since by afTuming a feries for x, 

 and making ufe of the multinomial theorem, the fame conelufion 

 will be derived by a procefs equally fliort. Yet it muft be ob- 

 ferved, that the multinomial theorem is only a particular theorem 

 far lefs extenfive indeed in its ufes than the method here given, 

 and not at all more ready in pradlice. 



Vol. VII. U u Example 



