120 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



~r \m -r /'2 ~ r ^'^ 



sin 2 V ^ 



S = + {p,'+p,^ - ^2 (,^2 + e^^)} '-^^^ - i (ei^^i + e^p^) vers 2^^P[ 



+ {;>3^-(-3 + fy}^^-4P3(e3 + f)vers2... 



kll8.) 



In order, however, to express this function S, as supposed by our general method, in 

 terms of the final and initial coordinates and of the time, we must employ the analo- 

 gous expressions for the constants p^ P2P3i deduced from the integrals (113.), namely, 



the following : 



jM, )ji — [x. e^ cos j«. t 

 Pi ~ sin ft i ' 



i"" >i2 — ft- gg COS ft t 



P2— sin j* ^ • 



Ps 



V ijg + Jl — (ve^ + ^ j cos V ^ 



sin v^ 



and then we find 



(119.) 



k5 — ^ 2 -h « . tan jt* / 



_4_ _L (^3 - ^sY 

 ' 2 • tanv/ 



- jw/ (^1 6?! + ??2 ^2) tan "V "■ " V3 + -f ) (^3 + ^) tan 



2 



^ . . . (120.) 



This principal function S satisfies the following pair of partial differential equations 

 of the first order, of the kind (86.), 





— 0I 



e 



3 ' 



(121.) 



J 



and if its form had been previously found, by the help of this pair, or in any other 

 way, the integrals of the equations of motion might {hy our general method) have been 

 deduced from it, under the forms, 



and 



^^1 = g^ = /M. (;ji — ej cotan^ ^ - i^^i tan -^, 

 tsTg = g^ = I"* (>?2 — ^2) cotan !«. # — ^ ^2 tan -^, 

 ^^3 = 8^ = K^3 - ^3) cotanv^ _ (^^gg + ^) 

 ^1= — 87- = |^(»7i — ei)cotan/A^ + ^;jitan -^, 

 ^2 = — 8l~ = i^ ('^2 — ^2) cotan ^ ^ + fA ??2 tan -g-, 



tan 



Li 

 2^ 



(122.) 



8S 



P3 = - g- 



(^3 — ^3) cotan c ^ + (v ;j3 + ^) tan ^ : , 



(123.) 



the last of these two sets of equations coinciding with the set (119.), or (113.), and 

 conducting, when combined with the first set, (122.), to the other former set of inte- 

 grals, (114.). 



