258 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAxMICS. 



2.m{x'la: + i/' lij + z' h z) = m^ (a/i § ^i + i/\ ^2/^ + z\ I z{) 1 



4-7^2(^2^-^2+^2^3/2+ ^2 ^^2) >. . (17.) 

 4-&C. + m (s/ Ix +1/ ly -\-z' lz)i \ 



^.m(dla-\-yih + die) =m^ {a\ la^-\- b\ Ih^-^ c\ I q) ") 



+ 7W2(a'2^«2 + ^'2^^2 + ^'2^^2) >• • (18.) 



-{- kc. -{-m (a! la -\-V Ih -\- d I c ). I 



Now X. being by supposition a function of the 3 n new marks of position ri^^ . . . ri^ , its 

 variation Ix.^ and its differential coefficient j/., may be thus expressed : 



hx. 



Zx. 



Ix. 



S^. = -^§;7i+,- ^^2 +... + , 



I 



'3-n» 



3n 



oo:. 



<- = .-^'''. + h'''^+---+ — "' 



3» 



(19.) 

 (20.) 



Zx. 



I 



and similarly for 3/. and z.. If, then, we consider x'. as a function, by (20.), of i^\... rl^ ^, 

 involving also in general yi^ . , . ri^^, and if we take its partial differential coefficients 

 of the first order with respect to n'l . . . rf^ ^, we find the relations, 



^x'. Ix. Zx^. Ix. Ix^. Ix. 

 I __ t . t I . » __ » 



8V, 



. (21.) 



3n 



3 71 



and therefore we obtain these new expressions for the variations Ix., ly., I z., 



1 





8V1 



Mi 



82'.^ 82'.^ 



+ 8V7^''3„. 



' 371 



sy. 



8V 



3 7» 



'3 7l' 



+ 



87^ ^''^»- 



3« 



(22.) 



Substituting these expressions (22.) for the variations in the sum (17.), we easily 

 transform it into the following, 



2 . m{x'lx+y'ly.^z'lz) =.2.?n {x' |^' + y |^' + z' I^J , $,,"j 

 + &c. + 2 . m fa:' ^ + y /4^ + z'^) .Ir 



> . (23.) 



8T 



ST 



8T 



- S7,^'?i +87;^''2+ . • • + 8^„ ^'^s,.' 



T being the same quantity as before, namely, the half of the final living force of the 



