296 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



V, = At-T.)<?';. . . (K^) 



though this is not, like (B^.)^ ^ perfectly rigorous expression for the remaining part 

 of the function. And in calculating this integral (K^.), for the improvement of an 

 approximation Vj, we may employ the following analogous approximations to the 

 rigorous formulae (D.) and (E.), 



and 



8V, __ 



(M5.) 



or with any other marks of final and initial position, (instead of rectangular coordi- 

 nates,) the following approximate forms of the rigorous equations (S.), 



together with the formula (M^.) ; by which new formulae the manner of motion of the 

 system is approximately though not rigorously expressed. 



It is easy to extend these remarks to problems of relative motion, and to show that 

 in such problems we have the rigorous transformation 



V,2=/'(T,-T„+T^)rf<, (05.) 



*^ 



and the approximate expression . 



V,2=/'(T,-T„)i<, (P^) 



V^i being any approximate value of the function V^ of relative motion, and V^ being 

 the correction of this value ; and T^^, T^g? l>eing homogeneous functions of the second 

 dimension, composed of the partial differential coefficients of these two parts V^i, V^2^ 

 in the same way as T^ is composed of the coefficients of the whole function V^. These 

 general remarks may usefully be illustrated by a particular but extensive application. 



Amplication of the foregoing method to the case of a Ternary or Multiple System, with 

 any laws of attraction or repulsion, and with one predominant mass. 



20. The value (68.), for the relative living force 2 T^ of a system, reduces itself 

 successively to the following parts, 2T/^\ 2T/^^, , . . 2T/"""^\ when we suppose that 



