The Dissolution of an Isotropic Solid. 155 



X, y co-ordinates of a point Q on the second curve, let PT 

 and PS be the normal and tangent at P, also let PQ = c be 

 a constant, then we have 



i^-xf^{Y-yf = <^ (i) 



c being constant, and all the other variables being regarded 

 as functions of X, we get by differentiation 



(X-.4-^).(V-,)(g-|) = (.) 



but \^ = tanPQR = tanPTS = cotPST = ; — %^ = -^ 

 X-x ^ tan PS 1 £Y 



dX 

 by substitution in (2) we get 



dx_^ f^_±\ 

 dX dVydX dXj 

 dX 

 .dY_^^^ (3) 



••dX~dX dx 

 dx 

 dX 

 Hence the tangent at Q is parallel to the tangent at P, and 

 PT is normal to the second curve at the point O. This 

 proposition will be of service in treating of the dissolution 

 of cylindrical solids, and surfaces of revolution. The co- 

 ordinates of the curves will be connected by the following 



relationship : 



x = X- CCOSCj) (4) 



J = Y - csin^ 



(p denoting the angle PTS. If for the angular functions we 



substitute their values in terms of the co-ordinates X, Y', 



we shall obtain equations which we may write 



.T=/,(X,Y/) (5) 



y=MX,Y,c) 



and from these we may obtain equations of the form 



Y = F,(^-j',^) (6) 



X = F2{xj',c) 



if the primitive equation be ^(X,Y) = 0, to obtain the de- 

 rived equations we must substitute for X and Y from (6). 



