SECT. I.] PARTICULAR SOLUTIONS. 215 



X 



initial values of u are actually expressed by a sin-; that is to \ 



say, that the primitive temperatures at the different points are 

 proportional to the sines of angles included between the radii Vv 

 which pass through those points and that which passes through 

 the origin, the movement of heat in the interior of the ring will 



Jet 



X 



be exactly represented by the equation u ae r * sin - , and if 

 we take account of the loss of heat through the surface, we find 



-(h + tyt . X 



v = ae v *- sm - . 

 r 



In the case in question, which is the simplest of all those which 

 we can imagine, the variable temperatures preserve their primi 

 tive ratios, and the temperature at any point diminishes accord 

 ing to the successive powers of a fraction which is the same for 

 every point. 



The same properties would be noticed if we supposed the 

 initial temperatures to be proportional to the sines of the double 



/Vl 



of the arc - ; and in general the same happens when the given 



n -v 



temperatures are represented by a sin , i being any integer 

 whatever. 



We should arrive at the same results on taking for the 

 particular value of u the quantity ae~ kn2t cos nx : here also we have 







2mrr = 2V, and n - ; hence the equation 



-k% ix 



u ae r cos 

 r 



expresses the movement of heat in the interior of the ring if the 



? 



initial temperatures are represented by cos . 



In all these cases, where the given temperatures are propor 

 tional to the sines or to the cosines of a multiple of the arc - , 



the ratios established between these temperatures exist con 

 tinually during the infinite time of the cooling. The same would 



