The Dissolution of an Isotropic Solid. 173 



I shall now consider the application of the foregoing 

 investigation to the solution of some of the more familiar 

 geometric forms. As a particular case consider the paral- 

 lelepiped, of which the lengths of the edges at any instant 

 are x, y, :y ; then the volume will be Ay::;, and after the 

 lapse of time dt the volume will become (x- 2 dx) (j/- 2dy) 

 {p- 2ds), if every face of the solid be equally acted upon by 

 the solvent ; also the area of the surface will be 2 {xy + sy + xa). 

 Neglecting products of small quantities we get the equation 



d7> = 2{xydz + xzdy + zydx) 

 and as the rate of action is everywhere the same 



dx = dy = dz 

 Integrating these equations, and denoting by z^^ y^, a^ 

 initial values we get the equations 



y ^Jo -Jo + ^, Z=^Z„-X^ + X. 



Writing /h for y„-\-So — 2x„, and /h for (jo — ,fj (So-Xo) the 



differential equations of solution becomes 



dx 



^ ~, 7 = - tdt 



jf + x'hi + x/i^ + r 



an expression which may be readily integrated, and its 



value determined at any time when the arithmetical values 



of the constants are assigned. If either y„=^o, or z„=:Xg, 



hi vanishes ; if both the equations are true h^ vanishes also. 



In this case the solid becomes a cube, and the integral 



becomes 



J 3 I 



c - r^n=Ao^-~^ '- + -— tan 



6 - x^^r v/3 ^iv/3^ 



the constant to be determined by the condition that when 

 /=() ,1'=^-" ; the time required for dissolution of the cube 

 may be obtained by writing o for x, and will be 



If r be negative, the relation between the length of the 

 edges of the cube and the time which has elapsed will be 



