CHAP. V. SECT. I.] PARTICULAR SOLUTIONS. 269 



time the differential equation, the condition relative to the 

 surface, and the initial state. It is easily seen that when these 

 three conditions are fulfilled, the solution is complete, and no 

 other can be found. 



284. Let y e mt u, u being a function of x, we have 



First, we notice that when the value of t becomes infinite, the 

 value of v must be nothing at all points, since the body is com 

 pletely cooled. Negative values only can therefore be taken for 

 m. Now k has a positive numerical value, hence we conclude 

 that the value of u is a circular function, which follows from the 

 known nature of the equation 



, &amp;lt;Fu 



mu = k -j-s . 

 dx 



Let u = A cos nx + B sin nx we have the condition m = k w 2 . 

 Thus we can express a particular value of v by the equation 



e -knH 



v = - (A cos nx -f B sin nx\ 

 so 



where n is any positive number, and A and B are constants. We 

 may remark, first, that the constant A ought to be nothing ; for 

 the value of v which expresses the temperature at the centre, 

 when we make x = 0, cannot be infinite ; hence the term A cos nx 

 should be omitted. 



Further, the number n cannot be taken arbitrarily. In fact, 

 if in the definite equation -j- + hv we substitute the value 



of v, we find 



nx cos nx + (hoc 1) sin nx = 0. 



As the equation ought to hold at the surface, we shall suppose 

 in it x = X the radius of the sphere, which gives 



Let X be the number 1 hX&amp;gt; and nX e, we have - - = X. 



tan e 



We must therefore find an arc 6, which divided by its tangent 



