140 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



If we choose to introduce into the expression (T2.), for $ Hg, the variation of the 

 time t, we have only to change Ir to Ir — lt, because, by (Q^.), 1 1 enters only so 

 accompanied ; that is, t enters only under the form t — r^, in the expressions of 

 l- J}. ^. ^. 7/'. z'. as functions of the time and of the elements ; we have, therefore, 



?^=_2!^'=-2.mp'; (211.) 



and since, by (H^.), (Q^.), 



Hi = 2.mp, (212.) 



we find finally, 



dt —"^ dt ^^^ '^ 



This remarkable form for the differential of H^, considered as a varying element, 

 is general for all problems of dynamics. It may be deduced by the general method 

 from the formulae of the 13th and 14th numbers, which give 



dt 5x, \ dri Sty 8«r 8)} / "^ " * ~^ Sxg^ V Sij Sot Sot Sij / I 



r (213.) 



Sxi S^ "*" Sxg 8^ "• * * * "*" Sxgn S^ S^ ' J 



«i »2 • • *6 » heing any 6 n elements of a system expressed as functions of the time and 

 of the quantities ;? tsr ; or more concisely by this special consideration, that Hj + Hg is 

 constant in the disturbed motion, and that in taking the first total differential coeffi- 

 cient of Hg with respect to the time, the elements may by (F^) be treated as constant. 

 It is also a remarkable corollary of the general principles just referred to, but one not 



Sx 

 difficult to verify, that the first partial differential coefficient ~ of any element «„ 



taken with respect to the time, may be expressed as a function of the elements alone, 

 not involving the time explicitly. 



On the essential distinction between the Si/stems of Varying Eleynents considered in this 

 Essay and those hitherto employed hy mathematicians. 



37. When we shall have integrated the differential equations of varying elements 

 (S2.), we can then calculate the varying relative coordinates i p? ^, for any binary sy- 

 stem (m, M), by the rules of undisturbed motion, as expressed by the equations (P.), 

 (Q2.), or by the following connected formulae : 



I = r (cos ^ 4- ~ sin (^ — v) sin v\ 

 r}=:r (sin ^ — — sin (^ ~ v) cos i'), 

 ? = ^ V2X» — X2 sin (^ - f) : 



(V2.) 



