of the Determinant al Relation of Dynamics. 533 



the points R, S having advanced beyond P, Q by an amount 

 (WK'=)8v.t. Hence 



CotRPQ = £*-;, 



and is proportional to t. As the time increases, the diagonal 

 P / S / becomes more and more elongated, but the area of the 

 parallelogram remains the same. 



Case II. Let the motion be simple harmonic, the accelera- 

 tion varying as the distance from a fixed point (fig. 2). 

 Then, by properly choosing the scale of representation of 



Fig. 2. 



--*" V 



R 



■"■"*-«.„ 



5 



• 







/ s 



if 



1 1 



1 1 



"**S» s 



"Or 



1* L 



i i 



vv 



\ 



IV 



i r 



/ 

 * 



velocity, the representative points of different particles will 

 all describe concentric circles about with uniform angular 

 velocity. Hence the figure PQRS will be brought into the 

 position P'Q'R'S' by rotating about through a certain 

 angle, and the areas of the two figures will of course be 

 equal. 



Case I. might be deduced as the limit of Case II. by 

 (i.) reducing the scale of representation of velocity so that 

 the circles become projected into ellipses ; (ii.) supposing the 

 centre to go off to infinity, so that these ellipses gradually 

 become elongated into parabolas. 



The case of a repulsive force varying directly as the distance 

 would be a little more complicated ; and it therefore seems 

 hardly worth while to give a proof for it, though the legiti- 

 macy of the corresponding inference for this case might be 

 inferred by means of " imaginary projection." The theorem 

 might possibly then be extended to the case of any variable 



