262 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



from the properties of our characteristic function. And if we were to suppose 

 (as it has often been thought convenient and even necessary to do,) that the n points 

 of a system are not entirely free, nor subject only to their own mutual attractions 

 or repulsions, but connected by any geometrical conditions, and influenced by any 

 foreign agencies, consistent with the law of conservation of living force ; so that the 

 number of independent marks of position should be now less numerous, and the force- 

 function U less simple than before ; it might still be proved, by a reasoning very simi- 

 lar to the foregoing, that on these suppositions also (which, however, the dynamical 

 spirit is tending more and more to exclude,) the accumulated living force or action 

 V of the system is a characteristic motion-function of the kind already explained ; 

 having the same law and formula of variation, which are susceptible of the same 

 transformations ; obliged to satisfy in the same way a final and an initial relation be- 

 tween its partial differential coefficients of the first order ; conducting, by the varia- 

 tion of one of these two relations, to the same canonical forms assigned by Lagrange 

 for the differential equations of motion ; and furnishing, on the same principles as 

 before, their intermediate and their final integrals. To those imaginable cases, indeed, 

 in which the law of living force no longer holds, our method also would not apply ; 

 but it appears to be the growing conviction of the persons who have meditated the 

 most profoundly on the mathematical dynamics of the universe, that these are cases 

 suggested by insufficient views of the mutual actions of body. 



9. It results from the foregoing remarks, that in order to apply our method of the 

 characteristic function to any problem of dynamics respecting any moving system, 

 the known law of living force is to be combined with our law of varying action ; and 

 that the general expression of this latter law is to be obtained in the following manner. 

 We are first to express the quantity T, namely, the half of the living force of the 

 system, as a function (which will always be homogeneous of the second dimension,) 

 of the differential coefficients or rates of increase 7j\, vi^, &c., of any rectangular co- 

 ordinates, or other marks of position of the system : we are next to take the variation 

 of this homogeneous function with respect to those rates of increase, and to change 

 the variations of those rates 1 7i\, I rl,^, &c., to the variations I ti^, ^ ^j^, &c., of the marks 

 of position themselves ; and then to subtract the initial from the final value of the 

 result, and to equate the remainder to ^ V — ^ § H. A slight consideration will show 

 that this general rule or process for obtaining the variation of the characteristic 

 function V, is applicable even when the marks of position pj^, rj.^, &c., are not all inde- 

 pendent of each other ; which will happen when they have been made, from any mo- 

 tive of convenience, more numerous than the rectangular coordinates of the several 

 points of the system. For if we suppose that the 3 n rectangular coordinates x^ y^ z^ 

 • • • *^n yn \' '^^^^ b^^" expressed by any transformation as functions of 3 n + A; other 

 marks of position, r,^ri2 . . - ri^^j^j^, which must therefore be connected by k equations 

 of condition. 



