resulting from Hamilton's Theory of Motion. 289 



for which the proposition can readily be verified, may first be 

 discussed. For this I select the motion of a pendulum which 

 takes place in the vertical plane of ocy about the downward- 

 directed axis of the positive y in infinitely small amplitudes; and 

 I give the determination of the two functions V and W accord- 

 ing to known methods *. 



The length of the pendulum being denoted by /, and the elon- 

 gation each time by 0, so that 



# = /sin#, y — l cos 0, 



the energy E expressed by the quantities p ( and q. becomes 



» 2 



?^T - 



where p=^^. Accordingly the differential equations of the 



motion are, taking account of the infinitely small amplitude, 



($__p_ 

 dt " I*' 



dt--2 W; 

 and the two integral equations 



P =Po cos a^JI t - 6 Q ls/Jl sin <\J 9 j t, 



where and p denote the values of 6 and p for / = 0. 



Introducing now these values into the expression for the vis 

 viva 



and substituting for the squares and products of the trigonome- 

 trical functions the doubled variables, we get 



4i* 4 Up 4 J Vi 



* Compare Hamilton's and Jacobi's examples. 

 Phil. Mag. S. 4. Vol. 48. No. 318. Oct. 1874. U 



