IS^G."] "Parameters to the Determination of certain Curves. 323 



I6a^x —Sax'' \ _ f\6 a'' x — S a x (X + x) 



-2a{X + x)"- + x(X + x)-j - \ + (X + a;)'- x 

 .: (X+ xf = 4Xx 

 .: (X - xf = 

 .*. X = X 



8 a X = 4 X* + (j/ - Y)-^ 

 ... (3^ _ Y) = 2 (2 a X - X"-)h 



(2a X- X')^ _ IX _ (2 o - X) 



X 



dX 



dY_ 

 dX 



\2a -X) 



(2aX-X^)^ 



The equation to a cycloid, equal and similar to the former, 

 having its origin at the extremity of the base instead of the 

 vertex, — a known property of the curve. 



On the above method of the involute, we may make the 

 following remarks. 



1. The involute is in all cases determinable in finite algebraical 

 terras when the evolute is rectifiable. 



2. Where the length of the evolute is expressed by a transcen- 

 dental function, the equation to the involute is always affected 

 by a similar function. 



Eocample 2. — To investigate formulae determining immediately 

 the involute to a caustic by refraction. 



Let S be the radiating point, and A P the refracting surface, 

 and S P, S P', consecutive rays, refracted so as to intersect in 



Q. Produce Q P, Q F, to M, M', so that P M = ^, P' M' = 



s P' 



— ; then since by the nature of caustics, we have ultimately 



y2 



