Arbitrary Constants in Mechanical Problems. 493 



I. -t — dt = (b 9 a) db + (c 9 a) dc-\-(e t a) de + &c. 



II. r/«= \b,a]^ dt + \c,a}^dt+\_e,a]~^ dt + &c. 



L J do L J dc L J ac 



with similar equations for the constants b, c, e, &c. (b, a), 

 (c,a) 9 (e, a), &c. \b, a\ [c, a], [e, a], &c. being the same as 

 in the notation of M. Poisson, Memoir es de V Institute 1816. 



III. That the quantities (&, a), (c, a), (e, a), &c. and also the 

 quantities [6, a], \_c, a], [e, a], &c. are constant. 



The well-known proof which M. Poisson has given of the 

 expressions which constitute the Unci theorem appears as 

 simple and direct as can be desired. See the TheorieAna- 

 lytique du Systeme du Monde, torn. L p. 222. I shall begin by 

 proving the 1st theorem. 



Considering R as a function of or, y, z, #',?/, z', 



dR jA dR dx .' dR dy lx dR dz Jj 



-dt = -v— -i—dt+ -j- ~-dt+ -f--r- dt 

 da da: da ay da dz da 



dR dx' ± dR dy f Jx dR dz' lA 



+ -tj i— di + -~n i dt + "37/ j- du 

 dr da dy da dz' da 



But if, as in the planetary theory, &c, R does not contain 



x, y\ d explicitly, 



dR dR _ dR _ 



dx l dy 1 " dz f 



dR lt dR dx Tj dR dy Jt dR dz 7 , ,■• „ 



— — dt = -5 -j- dt+-j -f- dt+- T -~ -?—dt (1.) 



da dx da dy da dz da x ' 



The following equations result from the usual conditions, 

 ... . dx dy dz . . . 



namely, that the expressions ttt* ~7n~ 9 ~j7 contlnue inform 



the same in disturbed as in the undisturbed motion. See the 

 Mecanique Anal., torn. i. p. 330. 



dx , dx ./" dx j dx , Q •■ ,« \ 



-=— da + — rr do +-r- dc + s— de + &c. = (2.) 



da do dc de v ' 



& daf Q; dH M&+Mde+kcSmO (3.) 



da db dc de v ' 



dz j dz ,, dz j dz , i ^ .. 



-j— da+- Tir db+- r —dc+ -j- d.e + 8cc. = (4.) 



da db dc de x ' 



Also, 

 rf^ _ dfa' ,, , rf^ , , dx 1 dR '. . 



