144 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



^ ^G , „ / 1 SG , 1 8G 1 8G 1 8G \ ,_,, . 



and each of the other analogous functions or combinations (Y^.) must satisfy an 

 analogous equation : if then we change G to Gj + Gg, and neglect the squares and 

 products of the coefficients of the small correction Gg, Gj being a first approximation 

 such as that already found, we are conducted, as a second approximation, on prin- 

 ciples already explained, to the following expression for this correction Gg : 



which may be continually and indefinitely improved by a repetition of the same pro- 

 cess of correction. We may therefore, theoretically, consider the problem as solved ; 

 but it must remain for future consideration, and perhaps for actual trial, to determine 

 which of all these various processes of successive and indefinite approximation, de- 

 duced in the present Essay and in the former, as corollaries of one general Method, 

 and as consequences of one central Idea, is best adapted for numeric application, and 

 for the mathematical study of phenomena. 



