SECT. II.] VALUE OF THE COEFFICIENTS. 253 



In general we could find each unknown by multiplying the 

 two members of each equation by the coefficient of the unknown 

 in that equation, and adding the products. Thus we arrive at the 

 following results : 



-ftf 2 sin I- 77 -fasm2-^&quot; +&c. = 2a i sin(i-l)l 



n n n ^ n 



2?r , 2?r - 2?r 2?r 



-- + GLCOS! +a 3 cos2 + &c.= 2 i cos(z-l)! 



n n n J n 







.2 +a 3 cos2.2 -f &c. = 2a, f cos (i-l)2 

 ?i ?i } n 



+ 2 sinl.3 + a 3 sin2.3 +&c.=Sosin(*-l)3 



71 71 ?i 



s^^cos 0.3 + 2 cosl.3 +CLCOS2.3 + &c. = 2a i cos(i-l)3^ 

 2 7i 71 n J n 



&c ............................................. . ..................... (M). 



To find the development indicated by the symbol %, \ve must 

 give to i its n successive values 1, 2, 3, 4, &c., and take the sum, 

 in which case we have in general 



n . ^ . ,. 1N/ . ., N 2?r , n ,-&amp;gt; , . - s , . . . 2?r 



g^=2asin(t-l)(;-l) and ^B =s2aodB(i-l)(;-l) . 



If we give to the integer^ all the successive values 1, 2, 3, 4, 

 &c. which it can take, the two formulae give our equations, and if 

 we develope the term under the sign 2, by giving to i its n values 

 1, 2, 3, ... n, we have the values of the unknowns A l9 J$ lt A 2 ,B Z , 

 A 3 , B 3 , &c.j and the equations (ra), Art. 267, are completely solved. 



272. &quot;We now substitute the known values of the coefficients 

 A lt B lt A 2 , B 2 , A 3 ,B S , &c., in equations (/A), Art. 266, and obtain 

 the following values : 



