128 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



(163.) 



But a little attention to the nature of this process shows that all the successive cor- - 

 rections to which it conducts can be only rational and integer and homogeneous 

 functions, of the second dimension, of the quantities Xj Xg X3 jt^ p2 Ps g, and that they 

 may all be put under the following form, which is therefore the form of their sum, 

 or of the whole sought function C ; 



C = (^-' a^ (Xi - p,Y + h^p^ (Xi - pi) + ^2 c^p2 



+ 1^"^ % {^2 - Vlf- + *^?2 (^2-^2) + ^^ <^f.V2 



+ ""^ «. ^2. - 7^3)^ + ^.Vz ih - Ps) + ^'^ c, p.^ 



+ f.g (h - Ps) + "^ ^.gPs + "^ ig^ 

 the coefficients a^ «, &c. being functions of the small quantities fju, v, and also of the 

 time, of which it remains to discover the forms. Denoting therefore their differen- 

 tials, taken with respect to the time, as follows, 



da^-= a!^dt, da^=z cH^dt, &c., (164.) 



and substituting the expression (163.) in the rigorous partial differential equation 

 (158.), we are conducted to the six following equations in ordinary differentials of 

 the first order : 



^2a',= (2«, + .2^)2; h\= i2a,-\-vU){h, + t); c', = i (5, -f- 0'; 1 



/',= (2«, + .2^) (/- i^2). h\=={h,+t){f,^^fi), z", =i(/ - J^2)2.|(165.) 



along with the 6 following conditions, to determine the 6 arbitrary constants intro- 

 duced by integration. 



t^ 



(166.) 



''^O — "~ 2 / 5 Oo — 2 ' -^O — 6 ' ^0 — 24 ' ^0 — "■ 24 5 «o — go- 



In this manner we find, without difficulty, observing that a^ h^ c^ may be formed 

 from «, h^ c^ by changing v to ^, 



a, = — iv^ t — \v cotan v t, o^ = — 2 l"*^ ^ ~~ 4 i"* cotan il t. 



&, = - / -h -;;- tan 2-, 

 / = J #2 _ _|_ _ cotan V t, 



b^= -t-\- — tQXi^, 



^f-— " 2/x2 + ^. 



1 ut 



3- tan — 



2 ' 



/8 I 



V 



2 



i = 27 — g;? — 27 cotan f /. 



(167.) 



The form of the function C is therefore entirely known, and we have for this func- 

 tion of elements the following rigorous expression. 



