472 On a Theorem in Plane Kinetic Trigonometry. 



of a material point moving in a plane it is easily worked out, 

 as 1 have found in endeavouring to write a continuation of 

 my article (Philosophical Magazine, October) on the Periodic 

 Motion of a Finite System, which I hope may be ready to 

 appear in the December number. Here is the theorem mean- 

 time. 



2. Let LABM, NBCQ, RCAS be three paths of a 

 particle moving freely in a plane, under influence of a force 



( - , — -= — J and projected from any three places in any 



direction in the plane, with such velocities that the sum of the 



kinetic and potential energies has the same value (E) in each 



case. The sum of the three angles A, B, C exceeds two right 



angles by an amount which, reckoned in radian, is equal to the 



surface-integral of V 2 log V(E — V), throughout the enclosed 



d 2 d 2 

 area ABC; V 2 denoting the Laplacian operation -7-^ + ~T~v 



3. To prove this ; remark that 



where yjr denotes any function of (<#, y) ; \^dxdy surface- 

 integration throughout any area ; j* ds line-integration all 



round its boundary ; and ~- rate of variation of ^ in the 



direction perpendicular to the boundary at any point. Hence 

 the surface-integral mentioned in § 2 is equal to 



f d ' 2(E~V)dn ••••••(!) 



—dY . 

 But — - - — is the normal-component force (N, we shall call 



it); and 2(E — V) is the square of the velocity (v 2 , we shall 

 call it). Hence (1) becomes 



; S*§ (2) 



But N/v 2 is the curvature (-, we shall call it), at any point in 



r 



any one of the three arcs A B, B C, C A. Hence, dividing 

 §ds into the three parts belonging respectively to these three 



