262 THEORY OF HEAT. [CHAP. IV. 



transmit heat, and wish to pass to the case of continuous bodies, 

 we must attribute to the coefficient k, which measures the yj^ocity 

 of transmission, a value proportional to the number of infinitely 

 small masses which compose the given body. 



3rd. If in the last equation which we obtained to express the 

 value of v or &amp;lt; (#, i), we suppose t = 0, the equation necessarily 

 represents the initial state, we have therefore in this way the 

 equation (p), which we obtained formerly in Art. 233, namely, 



+ sin as I f(x) sin x dx + sin 2# I f(x) sin 2# dx + &c. 



(*)&amp;lt;fo J J 



+ cos x \ f(x] cos xdx+ cos 2x I f(x) cos 2a? dx + &c. 



Thus the theorem which gives, between assigned limits, the 

 development of an arbitrary function in a series of sines or cosines 

 of multiple arcs is deduced from elementary rules of analysis. 

 Here we find the origin of the process which we employed to 

 make all the coefficients except one disappear by successive in 

 tegrations from the equation 



-f a^ sin x + a^ sin 2# + a z sin 3# + &c. 

 ^ * ~~ + b t cos x + 5 a cos 2x + b 3 cos 3# + &c. 



These integrations correspond to the elimination of the different 

 unknowns in equations (m), Arts. 267 and 271, and we see clearly 

 by the comparison of the two methods, that equation (B), Art. 279, 

 holds for all values of x included between and 2?r, without its 

 being established so as to apply to values of x which exceed those 

 limits. 



279. The function (x, t) which satisfies the conditions of 

 the problem, and whose value is determined by equation (E), 

 Art. 277, may be expressed as follows : 



+ {2sin3ic |^a/(a)sin3a4-2cos3^pa/(a)cos3a}e&quot; 32 ^-f- &c. 



