SECT. IV.] CONDITIONS OF DEVELOPMENT. 445 



order that the n equations may hold good together, that is to say 

 in order that equation (e) may be true when we give to x one of 

 these n values included between and X ; and since the number 

 n is infinite, it follows that the first member f (x) necessarily coin 

 cides with the second, when the value of x substituted in each 

 is included between and X. 



The foregoing proof applies not only to developments of the 

 form 



a sin jLs + sin x + a sin z# + . . . + a sin , 



it applies to all the functions &amp;lt; (frx) which might be substituted 

 for sin (/v&), maintaining the chief condition, namely, that the 



integral f dx $ (pp) $ (/A/C) has a nul value when i and j are 



Jo 

 different numbers. 



If it be proposed to develope/(#) under the form 



a, cos x a, cos 2j? a.cosix 



+7 O +.-.+ / + &C., 



b sm x 6 sm 2x b cos ix 



the quantities p lf /z 2 , ^ 3 ...^, &c. will be integers, and the con 

 dition 



I ec cos f2wt .] sin f 2?rj -^J = 0, 



always holding when the indices i and j are different numbers, we 

 obtain, by determining the coefficients a t , b iy the general equation 

 (II), page 206, which does not differ from equation (A) Art. 418. 



425. If in the second member of equation (e) we omitted one 

 or more terms which correspond to one or more roots /^ of the 

 equation (/), equation (e) would not in general be true. To 

 prove this, let us suppose a term containing /^ and a, not to be 

 written in the second member of equation (e), we might multiply 

 the n equations respectively by the factors 



dxsm(fijda:) 9 dxsmfajZdx), dx sin (//_. 3dar) . . . dx sin fondx) ; 



and adding them, the sum of all the terms of the second members 

 would be nothing, so that not one of the unknown coefficients 

 would remain. The result, formed of the sum of the first members, 



