SECT. II.] APPLICATION OF THE SOLUTION. 255 



in which only known quantities enter, namely, a v a 2 , a- 3 ... a n , 

 which are the initial temperatures, k the measure of the con- 

 ducibility, m the value of the mass, n the number of masses 

 heated, and t the time elapsed. 



From the foregoing analysis it follows, that if several equal 

 bodies n in number are arranged in a circle, and, having received 

 any initial temperatures, begin to communicate heat to each other 

 in the manner we have supposed ; the mass, of each body being 

 denoted by m, the time by t, and a certain constant coefficient by 

 k, the variable temperature of each mass, which must be a function 

 of the quantities t, m, and k, and of all the initial temperatures, 

 is given by the general equation (e). We first substitute instead 

 of j the number which indicates the place of the body whose 

 temperature we wish to ascertain, that is to say, 1 for the first 

 body, 2 for the second, &c.; then with respect to the letter i which 

 enters under the sign 2, we give to it the n successive values 

 1, 2, 3, ... n, and take the sum of all the terms. As to the 

 number of terms which enter into this equation, there must be 

 as many of them as there are different versed sines belonging to 

 the successive arcs 



0^,1^,2^3^ & 

 n n n n 



that is to say, whether the number n be equal to (2\ + 1) or 2\, 

 according as it is odd or even, the number of terms which enter 

 into the general equation is always \ + 1. 



274. To give an example of the application of this formula, 

 let us suppose that the first mass is the only one which at first 

 was heated, so that the initial temperatures a v a 2 , a 3 . . . a n are all 

 nul, except the first. It is evident that the quantity of heat 

 contained in the first mass is distributed gradually among all the 

 others. Hence the law of the communication of heat is expressed 

 by the equation 



1 2 



. = - a t -r - a. cos ( j 1) e m 



} n * n l J n 



2 2?T 



+ - a t cos (j 1) 2 e 



2 . 2?T -^ 



+ - j cos (j 1)3 e m n + &c. 



tv 7& 



