108 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



rig-orous and ordinary differential equations of the first order between those varying 

 elements and the time. Expressing, therefore, the quantity Hg as a function of these 

 latter variables, its variation I Hg takes this new form, 



m,=^^.^-^^^ + ^-^u, (57.) 



and gives, by comparison with the form (48.), and by (54.), 



yi — ^ LMii-^ Uh—y I3il^ r-Q\ 



8« — ^ • 8x S>, ' Sot — ^ • Sx 8w ' \^^-) 



'r 'r r r 



and thus the general equation (A^.) is transformed to the following, 



in which 



(8x. 8x 8x. 8x\ 

 j^j^~"fi~^i)'' (c^) 



so that it only remains to eliminate the variables tj -us from the expressions of these 

 latter coefficients. Now it is remarkable that this elimination removes the symbol t 

 also, and leaves the coefficients a^ ^ expressed as functions of the elements « alone, not 

 explicitly involving the time. This general theorem of dynamics, which is, perhaps, 

 a little more extensive than the analogous results discovered by Lagrange and by 

 PoissoN, since it does not limit the disturbing terms in the differential equations of mo- 

 tion to depend on the configuration only, may be investigated in the following way. 



15. The sign of summation 2 in (C^), like the same sign in those other analogous 

 equations in which it has already occurred without an index in this Essay, refers not 

 to the expressed indices, such as here f, .9, in the quantity to be summed, but to an 

 index which is not expressed, and which may be here called r ; so that if we intro- 

 duce for greater clearness this variable index and its limits, the expression (C^) be- 

 comes 



3n /8x. 8x^ 8x. 8x\ 



and its total differential coefficient, taken with respect to the time, may be separated 

 into the two following parts. 





> (60.) 



which we shall proceed to calculate separately, and then to add them together. By 

 the definition (54.), and the differential equations of disturbed motion (G.), 



dt 813- htivr 



snrj^c^.gj^ 8j^. S^^• /S_Hi 8_H,.-) 



^ (u)]l8i^8^^V8t3^-t- St^rJ ^^J^r\^\ ~^ ^\' r ^ ■^ 



