PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



277 



The expression (L^.) may therefore be thus transformed : 



^ V = X {(jc„ - «J (§ x„ - ^ a,;) + {i/„ - A J {^y,, - ^ ft J + {%,, - c,;) (^ z„ - h cj} 



»»■ rfr 



m, m, 



»*0 



(Q^) 



and may be resolved by our general method into twelve separate expressions for the 

 final and initial components of velocities, namely, 

 J_8V A^ 



1 8V 



x-, = 



3/1 = 



^1 = 



JO o ^^ 



^2 — 7»2Sy2 



X 



J_8V __ A 



tn^ 8 2^ TWj + 7^2 



1 SV A 



A 



W?! + /»2 

 A 



mcidj;^ 



(3/.-U + 



W2l + % 

 Wo 







and 



Zo = 



a. = 







A 



''l — m. ih. 



C, = 



«o = 



-^8V 

 ■-_l sv 



Wo Sflto 



5' ^ 2:2 8V ^ 



2 Wg 8^2 



(^^^ - O + ^7:;:^^ (f 872 + ^¥72/' 

 (3/.- U + ^7T^^ V^o 8^ - ^ 8j;;, 



(3/. - K) H- ^:5^ (^0 1^ - ^ Jj)> 



Wj + ?»2 

 A 



^1 + ^2^ " 



A 



A 



Wl + 7»2 

 A 



Co = 



-2 8V 



TTIq 8 ^2 ^'^i + ^2 



(^.. - ^./) + 



8r( 



wi + 7»2 \^<^s3; 



w, 



Wj + W2 



Usy-^iz)' 



besides the following expression for the time of motion of the system : 



_ 8V _ ^r dr 

 '-8H-/„ S' 



which gives by (K^.), and by (79.), (80.), 



(R2.) 



(S2.) 



(■P.) 



(W) 



The six equations (R2.) give the six intermediate integrals, and the six equations 

 (S2.) give the six final integrals of the six known differential equations of motion (74.) 

 for any binary system, if we eliminate or determine the three auxiliary quantities 



MDCCCXXXIV. 2 o 



