SECT. II.] FORM OF THE GENERAL SOLUTION. 247 



e 



sn 



2.1^ r + 7? 2 cos2.1^)/^ versinl ? 

 n n J 



9.TT 9.ir\ J*M verein 2 ^ 



n 



+ f^ 9 sin 2 . 2 + 3 cos 2 . 2 ) e~ - v 

 V n n y 



+ &c, 



f A n 27T A 27T) -=*&amp;lt; versin ** 



= j JjSin (n-1) + B i cos ( 1)0 \e m 



. 2 sin (n 1) 1 - H^ 2 cos (n 1)1 \ e m * 



&c 



To form these equations, we must continue in each equation 



the succession of terms which contain versin , versin 1 , 



n n 



versin 2 , &c. until we have included every different versed 



sine ; and we must oniit all the subsequent terms, commencing 

 with that in which a versed sine appears equal to one of the 

 preceding. 



The number of these equations is n. If n is an even number 

 equal to 2t, the number of terms of each- equation is i + 1 ; if n 

 the number of equations is an odd number represented by 2/+ 1, 

 the number of terms is still equal to i + I. Lastly, among the 

 quantities A I} B lt A 2 , B^ &c., which enter into these equations, 

 there are some which must be omitted because they disappear of 

 themselves, being multiplied by nul sines. 



267. To determine the quantities A V B^A V B V .A^B V &c., 

 which enter into the preceding equations, we must consider the 

 initial state which is known : suppose t = 0, and instead of 

 a lt 2 , 3 , &c., write the given quantities a x , a 2 , a 3 , &c., which are 

 the initial values of the temperatures. We have then to determine 

 A lf B lt A 9 , B 2 , A a , B 3 , &c., the following equations: 



