(153.) 



126 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



C = ^f^ (V ^2 _ ^2 + y^l t2 _ ^2)dt 



and has therefore the following for a first approximate value, obtained by treating 

 the elements k^ »2 «3 ^i ^2 ^3 ^^ constant and equal to their initial values e^ e^ e^ p^ p^ P-i-> 



C = - -I" 1^2 (e^2 + ei) + ^2 63^ I + I 1^2 (^^2 4. p2) J^ ,2^^2| 





(154.) 



In like manner we have, as first approximations, of the kind expressed by the ge- 

 neral formula (Z^), the following results deduced from the equations (151.), 



^2=P2- f^^ (^2 ^ + "i P2 ^> 

 and therefore, as approximations of the same kind. 



(155.) 



e^ = 



Pi* 



^2 — ^ — Q P2* 



>^\-Pl 



^3 = ~^/>3^ + -6^^2--Vf. 



(156.) 



Substituting these values for the initial constants e^ 63 e^ in the approximate value 

 (154.) for the function of elements C, we obtain the following approximate expres- 

 sion Cj for that function, of the form supposed by our theory : 



r _ ^ f (^1 -Pif + {\ -P^r , (^3 - Psf \ 



-^{{^^l- Pl)Pl-^ {^2-P2)P2 + {^3-P'6) (Pi- ^ g*)] > (157.) 



The rigorous function C must satisfy, in the present question, by the principles of the 

 eighteenth number, the partial differential equation, 



-I7 = |-{(u; + ^i0 +Wi + h')} + ^{^ + ht-^gfi) ;(158.) 

 and if it be put under the form (U^), 



C = Ci -I- C2, 

 Ci being a first approximation, supposed to vanish with the time, then the correction 

 C2 must satisfy rigorously the condition 



