SECT. II.] LATER TEMPERATURES. 257 



considerable, the value of a,- is represented without sensible error 

 by the equation, 



1 2 f 2?r 2?r 



a,- = - 2 t a i 4- - -tain (j -1) 2a f - sin (i 1) 

 n n { n n 



4 cos ( j - 1) 2a,- cos (i - 1) 

 ft ?? ^ 



Denoting by a and 6 the coefficients of sin ( / - 1) and of 



n 



cos (j 1) , and the fraction e~~&amp;gt;* m &quot;*&quot; by G&amp;gt;, we have 



7i 



1 ( 2-7T 9^ 



o ; = - 2 4 4- to sin (j - 1) 4 6 cos (j - 1) ~ 

 w ( n n 



The quantities a and b are constant, that is to say, independent 

 of the time and of the letter j which indicates the order of the 

 mass whose variable temperature is a,-. These quantities are the 

 same for all the masses. The difference of the variable tempera 

 ture a.j from, the final temperature - 2a f decreases therefore for 



IV 



each of the masses, in proportion to the successive powers of the 

 fraction &&amp;gt;. Each of the bodies tends more and more to acquire 



the final temperature - 2 a it and the difference between that 



final limit and the variable temperature of the same body ends 

 always by decreasing according to the successive powers of a 

 fraction. This fraction is the same, whatever be the body whose 

 changes of temperature are considered ; the coefficient of co* or 



(a sin Uj 4 & cos HJ), denoting by KJ the arc ( j - 1) - , may be put 



under the form A sin (uj 4- B), taking A and B so as to have 

 a = A cos B, and b = A sin B. If we wish to determine the 

 coefficient of to* with regard to the successive bodies whose 

 temperature is a j+l) a j+2) a j+3&amp;gt; &c., we must add to HJ the arc 



- or 2 , and so on ; so that we have the equations 

 n n 



% - - 20; = A sin (B 4 %) to* + &c. 

 n 



OLJ . , - - 2a f - = A sin [B 4 Uj 4 1 J at 4- &c. 

 n \ n / 



F. H. 17 



