PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



287 



component accelerations of the three points m^ m<^ m^, or the second differential co- 

 efficients of their coordinates, taken with respect to the time. To integrate these 

 equations is to assign, by their means, nine relations between the time t, the three 

 masses m^ m^ Wg, the nine varying coordinates x^ y^ z^ x<i y^ ^2 -^3 Vz Hi ^^^ their nine 

 initial values and nine initial rates of increase, which may be thus denoted, a^ h^ Cy 

 a^ h^ C2 a^ Z>3 C3 a\ b\ c'j a'2 b'2 (^2 ^'3 ^'3 ^'3- The known intermediate integral con- 

 taining the law of living force, namely, 



^ m, w + y 1' + ^?) + Y ^2 W + y7 + ^7) + ^^3 W + y7 + ^'3') 



gives the following initial relation : 



(124.) 



1^\ »*2/o^'' ^^ + '^\ ^3/0^^' ^^ + ^2 ^3/0^^' ^^ + H, 



- (125.) 



(1,2) /.(I, 3) /.(2, 3) 



in which f^ ' , /o ' > fo ' ) ^^'^ composed of the initial coordinates, in the same 

 manner as f^^' f^^' ^^ f^^' ^^ are composed of the final coordinates. If then we knew 

 the nine final integrals of the equations of motion of this ternary system, and com- 

 bined them with the initial form (125.) of the law of living force, we should have ten 

 relations to determine the ten quantities t a\ b\ c\ d^ h\ c'2 a'3 h\ c'3, namely, the time 

 and the nine initial components of the velocities of the three points, as functions of 

 the nine final and nine initial coordinates, and of the quantity H, involving also the 

 masses ; we could therefore determine whatever else depends on the manner and time 

 of motion of the system, from its initial to its final position, as a function of the same 

 extreme coordinates, and of H. In particular, we could determine the action V, or 

 the accumulated living force of the system, namely, 



V = m,f^ (^,2 + y,2 + si',2) d t + in,J^ {^i + y,2 + ^t) d t ] 



+ «^iJ^W+l/z^ + ^z^)dt, 



J 



(A^) 



as a function of these nineteen quantities, ^1 3/1 ^1 X2 3/2 ^2 ^i Vz % ^1 ^1 ^1 «2 ^2 ^2 

 a^ ^3 C3 H ; and might then calculate the variation of this function, 



).T7 ^^ ^ .^V). .^V. ,8V. ,8V SV 



8V ,§V. 8V. . 8V. , SV., , 8V 



^Xa 





+ 8^^^3 + g^^^3 + 8^5r3 



(BK) 



2 p2 



