324 Mr. Museley on a new Application of the Method of [May, 

 ... ^ + P Q = ^ + F Q 



m w 



Or P M + P Q = P' M' + P' Q 

 ... QM = QM' 



and the same being true for all positions of M similarly deter- 

 mined, the locus of M is the involute to the locus of Q, i. e. to 

 the caustic. 



IS'ow to determine the locus of M, we have 



(X _,)^ + (Y-2/r=-^^' (1) 



and differentiatine; v ith respect to .r and y. Since the position 

 (of the normal P M and the point M, and /.) of the tangent at M 

 is a function of the co-ordinates of P, we set 



Eliminating x and y between the Equations (1) and (2), and 

 the given equation to the refracting surface, we have the 

 required equations in X and Y. 

 Thus when the refraction is made 

 at a plane surface A P ; taking A for 

 the origin, if A S = a, we have 



Ml- 



(ni" - 1)" 



...Y»- = K-i)Sx"-;j; 



the equation to an ellipse or hyperbola, according as (/«) is less 

 or greater than unity. 



When m equals unity, the general equations become 



(X -xY +(y -^yr = ^^' +f- 



X + \ 7- = 



Or 



X,^' + Y'^ - 2 (X .r + Y y) = (1) 



XeZi- + Yrfj/= (2) 



