582 BEPOKT— 1888. 



(in which p denotes the perpendicular from the common centre on the tangent, 

 V the velocity in the orbit, T the tension of the string, and h, 0, and A constants), 

 together with the additional formula (proved in my edition of 'Todhunter's 

 Statics'). 



F = C dp 

 p- dr 

 we have 



— oc — CKZ V oc 1 ecu. 

 Y p '^ 



If, instead of supposing the three curves identical, we suppose them similar and 

 similarly placed with respect to the centres in question, the above results will 

 obviously remain true. 



Further, instead of supposing the forces central, and the index a function of 

 distance from centre, let us make the more general supposition that, at corre- 

 sponding points of the orbit and the string, the forces P and F are parallel and 

 opposite, and that, at the corresponding point of the curved ray, the direction of 

 most rapid increase of /x is the same as the direction of P ; then we have the fol- 

 lowing equations, in which cj) denotes the angle between the direction of F and the 

 tangent drawn in the direction of increasing s. 



P cos (^ = - A ^' (1) Pam<p ='^. . . . (2) 



as Z p 



Fco3<^= -^ (3) Fsin^=^ . . . . (4) 



Wo p 



^&Jicos</> = '^°^-'^ ... (5) - ^^^sin,^ = - . (6) 

 dr ^ ds dr ^ p ^ 



From (1) and (2), cot (/, = - p i}Pll ; 

 „ (3) and (4), cot (^ = ' P^dr '^ 



„ (5) and (6), cot (^ = - p IMi". 



Equating the three values of cot (j), we have 



d log V d log T 

 ds ~ ds 



Hence the logarithms of v, T and p, change by equal amounts ; that is, 



U oc T oc /Lt. 



From (2) and (4), - = ^. But ^ is constant ; 



p 



.*. — oc U oc T oc u. 

 F 



The proportionality of v, T and p can be proved without the calculus by re- 

 garding the three curves as the limit of three similar polygons. 



In the first polygon a particle moves along the sides under no forces, but is 

 acted on by impulsive forces at the corners. 



The second polygon is a polygon of string kept tight by forces applied at the 

 corners, the lines of action of these applied forces being parallel to the impulses in 

 the first polygon, but opposite to them in direction. 



The third polygon is the path of a ray which undergoes refraction at each 

 corner by passing out of one uniform medium into another uniform medium ; and 



