444 THEORY OF HEAT. [CHAP. IX. 



equal to dx, so that we have ndx = X, and writing f (x) instead of 

 xF(x),wQ denote by /^/^jf. .../;.../, the values of /(#), which 

 correspond to the values dx, 2dx, Sdx, . . . idx . . . ndx, assigned to 

 x ; we make up the general equation (e) out of a number n of 

 terms; so that n unknown coefficients enter into it, a v a 2 , 3 , ... 

 ^...a^ This arranged, the equation (e) represents n equations 

 of the first degree, which we should form by substituting succes 

 sively for x, its n values dx, 2dx, 3dx,...ndx. This system of n 

 equations contains yj in the first equation, / 2 in the second, / 3 in 

 the third, f n in the n ih . To determine the first coefficient a lt we 

 multiply the first equation by a- lt the second by cr 2 , the third by 

 &amp;lt;7 3 , and so on, and add together the equations thus multiplied. 

 The factors &amp;lt;7 1} cr 2 , o- g , ...o- tt must be determined by the condition, 

 that the sum of all the terms of the second members which contain 

 a a must be nothing, and that the same shall be the case with the 

 following coefficients a a , c& 4 , ...a n . All the equations being then 

 added, the coefficient a^ enters only into the result, and we have 

 an equation for determining this coefficient. We then multiply 

 all the equations anew by other factors p l , p 2 , p 3 ,...p n respectively, 

 and determine these factors so that on adding the n equations, all 

 the coefficients may be eliminated, except a 2 . We have then an 

 equation to determine a 2 . Similar operations are continued, and 

 choosing always new factors, we successively determine all the 

 unknown coefficients. Now it is evident that this process of elimi 

 nation is exactly that which results from integration between the 

 limits and X. The series &amp;lt;r l , cr 2 , &amp;lt;r 3 ,...&amp;lt;r n of the first factors is 

 dx sin (fijdx), dx sin (p^dx), dx sin (pfidx) ...dx sin (^ndx). In 

 general the series of factors which serves to eliminate all the co 

 efficients except a it is dx sin (^dx), dx sin (&amp;gt;. 2dr), dx sin (^ 3dx) . . . 

 dx sin (pjridx) ; it is represented by the general term dx sin (^x), 

 in which we give successively to x all the values 



dx, 2f&, %dx, . . . ndx. 



We see by this that the process which serves to determine these 

 coefficients, differs in no respect from the ordinary process of elimi 

 nation in equations of the first degree. The number n of equations 

 is equal to that of the unknown quantities a lf 2 , a a ...a n , and is 

 the same as the number of given quantities /,,/,,/,... /^ The 

 values found for the coefficients are those which must exist in 



