118 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS, 



and 



^1 ^ '^'l (^> ^U ^2? *3J ^1> ^2j ^3)^ 1 



^2 = "^2 (^^ «1J «2J «3J ^U ^^2J ^3)^ )• (104.) 



^3 = '^'S (^5 ^15 ^2> *3J ^15 ^2' ^3) 5 J 



and may call the quantities »i «2 ^3 ^1 ^2 ^3 ^be 6 varying elements of the motion. To 

 determine these six varying- elements, we may employ the six following- rigorous equa- 

 tions in ordinary differentials of the first order, in which H2 is supposed to have been 

 expressed by (103.) and (104.) as a function of the elements and of the time : 



^ Hg -^ 



8A3V 



y (105.) 



dx^ 8 Ho I 



dt ~~ 8x3 ' J 



and the rigorous integrals of these 6 equations may be expressed in the following 

 manner, 



_ 8^ __ 8_E _ 8^ 1 



^1 — 8X1' ^— 8X2' ^3 — gxg' I 



V\~ — le^' P2— — le^' P^ "~ "" 8^3' J 



the constants e^ e^ e^p^ P2P3 retaining their recent meanings, and being therefore the 

 initial values of the elements «j «2 ^2 ^1 ^2 ^3 ? while the function E, which may be 

 called the function of elements, because its form determines the laws of their variations, 

 is the definite integral 



E=/'(x,^ + K,^ + X3^^-H,)rf*,. ..... (.07.) 



considered as depending on x^ n^ H ^1 ^2 ^3 ^iid/. The integrals of the equations (105.) 

 may also be expressed in this other way. 



^1 ~ + 8 Ai' *2 — + g A2' ^3 — + 8 y^ 



> (108.) 



_ _ 8C _ ic _ i^ 

 C being the definite integral 



^, /.</ 8H2 , 8H2 , 8H2 tt\,^ .,^^x 



^^VoV^iT^ + ^^T^ + ^^Tlif-H^)^^, (109.) 



regarded as a function of \ Xg ^.3 p^ p^ p^ and ^ : and it is easy to prove that each of 

 these two functions of elements, C and E, must satisfy a partial differential equation 

 of the first order, which can be previously assigned, and which may assist in disco- 

 vering the forms of these two functions, and especially in improving an approximate 

 expression for either. All these results for the motion of a single point, are analogous 

 to the results already deduced in this Essay, for an attracting or repelling system. 



