SECT. II.] REMARKS. 261 



and representing the quantity gir by k, we have 



= g \f(x)dx+ ( sin x \f(x) sin xdx + cos x I /(a;) cos #cta J e w 



+ (sin 20ma) sin 2#efo+cos2# //(#) cos 2# dxj e~^ kt 



TTV 



+ &c. 



278. This solution is the same as that which was given in the 

 preceding section, Art. 241 ; it gives rise to several remarks. 1st. 

 It is not necessary to resort to the analysis of partial differential 

 equations in order to obtain the general equation which expresses 

 the movement of heat in a ring. The problem may be solved for f 

 a definite number of bodies, and that number may then be sup- \ 

 posed infinite. This method has a clearness peculiar to itself, and 

 guides our first researches. It is eas^afterwards to pass to a 

 more concise method by a process indicated naturally. We see 

 that the discrimination of the particular values, which, satisfying 

 the partial differential equation, compose the general value, is 

 derived from the known rule for the integration of linear differ 

 ential equations whose coefficients are constant. The discrimina 

 tion is moreover founded, as we have seen above, on the physical 

 conditions of the problem. 2nd. To pass from the case of separate 

 masses to that of a continuous body, we supposed the coefficient Jc 

 to be increased in proportion to n, the number of masses. This 

 continual change of the number k follows from what we have 

 formerly proved, namely, that the quantity of heat which flows 

 between two layers of the same prism is proportional to the value 



of y- , x denoting the abscissa which corresponds to the section, 



and v the temperature. If, indeed, we did not suppose the co 

 efficient k to increase in proportion to the number of masses, but 

 were to retain a constant value for that coefficient, we should 

 find, on making n infinite, a result contrary to that which is 

 observed in continuous bodies. The diffusion of heat would be 

 infinitely slow, and in whatever manner the mass was heated, the 

 temperature at a point would suffer no sensible change during 

 a finite time, which is contrary to fact. Whenever we resort to 

 the consideration of an infinite number of separate masses which 



