TJie Dissobitio7i of an Isotropic Solid. 165 



of ,1', then the instantaneous curve will trace out the surface 

 bounding the undissolved portion at any instant, when a 

 prolate spheroid is acted upon by a solvent ; the values of 

 X and y substituted in the formula irj y^dx will give the 

 volume of this undissolved solid ; in like manner, if the 

 figure revolve round the axis of y, from the foregoing in- 

 vestigation we may deduce the value of the integral irj x~dy, 

 being the volume at any instant of the solid generated by 

 the solution of an oblate spheroid. 



The second proposition, before referred to, which enables 

 us to investigate geometrically the dissolution of all solids is 

 as follows. // a surface be drawn cutting the lines normal 

 to a given surface at a constant distance from the surface, then 

 these lines will be normal to the surface so drawn. Let 

 X, Y, Z, be co-ordinates of a point on the given surface 

 0(X, Y, Z,) = and ,r, y, z co-ordinates of a point on the 

 instantaneous surface ;//(.r, J', r) = 0, then c denoting a con- 

 stant we have the equation 



(X-.rr + (Y-jO^ + (Z-s^) = .l (18) 



At a contiguous point we shall have 



(X + ^X - X - dxf + {Y + dY-y- dyf + {Z + dZ-z- dzf = c'^ ; 

 expanding and eliminating the constant, we get 



(X - .t)(^ - ^v) + (Y -y){dY -dy) + {Z- z){dZ - dz) 



+ (^X - dxf + (^Y - dyf + {dZ - dzf = 0; 



if the second point be taken indefinitely near to the first 

 point, then neglecting small quantities of the second order, 

 the last equation may be written in the form 



(X - .v)^X + (Y -y)dY + (Z - z)dZ = (X - x)dx 



+ (Y -y)dy + (Z - z)dz. 



since the line is normal to the surface 0(X,Y,Z) the ex- 

 pression on the left vanishes. Hence we have 

 (X - x)dx + (Y -y)dy + {Z- z)dz = 0, 

 and this is the condition to be fulfilled, that the line may be 

 a normal to the instantaneous surface (p(x,y,z). 



