264 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



Comparing the two expressions (33.) and (34.), we find by (36.) the relations 



ST _\ H? . . / ST \ SrJ/^ 1 



ST \ H 



8T /5T\ , / ST \ ii, , /_1T_\ H. , , /_iT_\y:^ 



ST /STx . / ST \H, , / ST vg4>, . / ^T \H, 



W, = Wj + Wsn + J ^^^ "^ V^V3„ , J S>,, "^ • • • -^ VSV3„+J S,,' 



K37.) 



^T 

 S>3 



T /ST\ , / ST \ Sri/, / ST_\ M^ / ST \H^; 



^"VV3j + VSV3„ + Js,3„+VSV3„ + JS,3„ + ----f VSV3„ + ./S>,3n 



which give, by (32.), 



ST. , ST. , . ^T V _ 



'ST> 



SV 



3n 



'3 re 



(S) ^"1 + (© ^"^ • • • + (S^j ^''3» + .; 



} 



(38.) 



we may therefore put the expression (Q.) under the following more general form, 

 'ST\ V ^ /ST, 



hY = 2.(jj)^,'^t.(j^)le+an, 



(AK) 



'ST 



the coefficients (yr) being formed by treating all the S n -\- k quantities fj\, rj^, . . . 



rl^ „ J. ;fc5 as independent ; which was the extension above announced, of the rule for 

 forming the variation of the characteristic function V. 



We cannot, however, immediately decompose this new expression (A\) for l V, as 

 we did the expression (Q.), by treating all the variations Iri,^ e, as independent ; but 

 we may decompose it so, if we previously combine it with the final equations of con- 

 dition (31.), and with the analogous initial equations of condition, namely. 



= 02(61, e2,...e3„_^,). 



> 







O^(e„eo 



(39.) 



Ak. 



which we may do by adding the variations of the connecting functions <pi, 

 Oj, . . . 0^, multiplied respectively by factors to be determined, Xj, . . . X^, A^, 

 In this manner the law of varying action takes this new form, 



and decomposes itself into Qn-\-2 k -{- 1 separate expressions, for the partial differ- 

 ential coefficients of the first order of the characteristic function V, namely, into the 

 following. 



