248 THEORY OF HEAT. [CHAP. IV. 



a x =A 1 sin 0.0^&quot;+ A 9 sin 0.1 + A sin 0.2 + &c. 

 n w ?& 



+ B. cos . + jR, cos . 1 + J5_ cos . 2 + &c. 

 ?i w n 



t . - _ 2?r , - ^ 2?r . ._ 2?r n 



2 = A 1 sin 1 . + A sin 1 . 1 + A. sin 1 . 2 + &c. 

 n n n 



+ &ooai .0 + # 2 cos 1 . 1 + K cos 1 . 2 + &c. 



n n n 



8 = A l sin 2 . 2 - + 4 a sin 2 . 1 + A 8 sin 2 . 2 + &c. 



+ A cos 2 . + B. 2 cos 2 . 1 + K cos 2 . 2 + &c. 

 n n n 



/7T 



w - 1) - 



268. In these equations, whose number is ??, the unknown 

 quantities are A lt B lt A 2 , B 2 , A 5 , B s , &c., and it is required to 

 effect the eliminations and to find the values of these unknowns. 

 We may remark, first, that the same unknown has a different 

 multiplier in each equation, and that the succession of multipliers 

 composes a recurring series. In fact this succession is that of the 

 sines of arcs increasing in arithmetic progression, or of the cosines 

 of the same arcs ; it may be represented by 



sin Qu, sin lu, sin 2w, sin 3w, ... sin (n 1) u, 

 or by cos Qu, cos lu, cos 2w, cos Su, ... cos (n I) u. 



/2?r\ 

 The arc u is equal to i I j if the unknown in question is A. +l 



or B. +1 . This arranged, to determine the unknown A i+l by means 

 of the preceding equations, we must combine the succession of 

 equations with the series of multipliers, sin Ow, sin lu, sin 2u, 

 sin Su, ... sin (n l)u t and multiply each equation by the cor 

 responding term of the series. If we take the sum of the equa- 



