PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



261 



and therefore that the total variation of the new partial differential equation (T.) may 

 be thus written, 



2(VJ^-^5,) = 2.^S, + SH: (V.) 



in which, if we observe tliat rj = -jj, and that the quantities of the form n are the 

 only ones which vary with the time, we shall see that 



2.;;^^=2(^^-^.§, + ^-g^.^ej+-^gjj.m, . . (29.) 

 because the identical equation I dV =^ dlY gives, when developed. 



= 2(rf^.B, + ^^.^e)+/g^.m. I 



(30.) 



Decomposing, therefore, the expression (V.), for the variation of half the living force, 

 into as many separate equations as it contains independent variations, we obtain, not 

 onlv the equation 



_d_lX — ^ 



dtm — ^' (K.) 



which had already presented itself, and the group 



d 8V 



d 8V 



dt 8^1 — ^' dt S^2 — ^^ • • • 



d 8 V 



dt ^e, 



Sn 



= 0, 



(W.) 



which might have been at once obtained by differentiation from the final integrals (S.), 

 but also a group of 3 w other equations of the form 



_^ 8V __ 8T _ 8U 



dt ^r^ 8,j ~ 8)j» V^'J 



which give, by the intermediate integrals (R.), 



d IT 8T_ SU 



rf/ 8>}' ""17— 8), ' (Y.) 



that is, more fully, 



A. LI 



dt 8)j'i 



d 8T 

 dt 8 )j'2 ■ 



d 8T 



3n 



8T 



8)3i 



8T 



8T 



8U 

 8>)i 



8U 

 8)jJ 



"I 



8U 



(Z.) 



These last transformations of the differential equations of motion of the second 

 order, of an attracting or repelling system, coincide in all respects (a slight dif- 

 ference of notation excepted,) with the elegant canonical forms in the Mdcanique 

 Analytique of Lagrange ; but it seemed worth while to deduce them here anew, 



MDCCCXXXIV. 2 M 



