854 Mr. J. Rose-Innes on the Motion of a Pendulum 



the side BC to meet the polar circle of B, and let the point 

 of intersection be called E. Then since 



cos AC X cos BC = cos AB, a constant, 



we have also 



sin CD x sin CE =a constant. 



This is the well-known property o£ a sphero-conic with 

 regard to its two cyclic arcs. (See Salmon's ' Solid 

 Geometry/) Hence we see that the required locus is a 

 sphero-conic which passes through the poles of its two cyclic 

 arcs. 



Relations between the Elliptic Functions of any argument u and 

 those o/K — u. 



Let ABC as before be a right-angled triangle with the 

 right angle at C. Produce the sides AB and AC until they 

 meet the polar circle of A in the points D and E respectively ; 

 produce the sides BA and BC until they meet the polar circle 



of B in the points F and G respectively. Let the oval curve 

 denote the locus of C, and let the intersection of the two 

 polar circles which lies on the same side of AB as C be 

 denoted by H. 



We see that the area of the quadrilateral CEHG remains 

 constant as C moves, since each of the four angles contained 

 in it remains separately constant. Subtract this quadrilateral 

 and half the sphero-conic from the triangle HFD ; the 

 remainder consists of two curvilinear quadrilaterals, whose 

 sum is therefore proved independent of the position of C. 

 By taking the particular case when A is a right angle and B 

 is zero, we see that the value of this constant sum is equal to 



