558 Mb. stokes, on THE CRITICAL VALUES OF 



where sines and cosines occur alternately, and two signs of the same kind are always followed by 

 two of the opposite. The expression for <p'^(fi) when n is odd might be found in a similar manner. 

 These formulae enable us to express f^ (x) when (p{^) is an arbitrary function which has to be 

 determined, and/(0), &c. are given. 



30. If however d)(/3) should be given, and /(O), &c. be unknown, ^{/3) will admit of expansion 

 according to powers of /3"', beginning with the first, provided we treat sin /3a or cos /3a as if it 

 were a constant coefficient ; and sin /3a, cos /3a will occur with even and odd powers of /3 respec- 

 tively. The possibilitv of the expansion of (^(/3) in this form depends of course on the circum- 

 stance that (p{x) is a function of the class which it is proposed in Art. 1, to consider, or at least 

 with the extension mentioned in Art. 23. It appears from (55) that in order to express fix) as a 

 definite integral of the form (50) we have only got to expand (p (0), to differentiate (50) /u times 

 with respect to x, differentiating under the integral sign, and to reject those terms which appear 

 under the integral sign with positive powers of /3 or with the power 0. The same rule applies 

 whether ii be odd or even. 



31. If we have given <j)(a), but are not able to evaluate the integral (50), we may notwith- 

 standing that find the values of .r which render /{x) or any of its derivatives discontinuous, and 

 the quantities by which the function considered is suddenly increased. For this purpose it is only 

 necessary to compare the expansion of (p(,(i) with the expansion 



(P{fi) = ~ SQcos lia - -^^ SQ, sin ^a- (56), 



irp 7r/5 



given by (55), just as in the case of series. 



We may easily if we please clear the function (^(/3) of the part for which f(a:) or any one of 

 its derivatives is discontinuous, or does not vanish for .r = and .v = a. For this purpose it will be 

 sufficient to take any function F{w) at pleasure, which as well as its derivatives of the orders 

 considered has got the same discontinuity as/(a') and its derivatives, to develope F{v) in a definite 



integral of the form / 4>(/3) sin (3xd^ by the formula (51), and to subtract F(.r) from /(,r) and 

 •'o 



<l>(/3) from d)(/3). It will be convenient to choose such simple functions as / + m<v + tix- ; 



/ sin a; -I- »B cos ,x' ; le'' + me~°', &c. for the algebraical expressions of i^(.r) for the several 



intervals throughout which it is continuous, the functions chosen being such as admit of easy 



integration when multiplied by sin (ixd.r, and which furnish a sufficient number of indeterminate 



coefficients to allow of the requisite conditions as to discontinuity being satisfied. These conditions 



are that the several values of Q, Q„ &c. shall be the same for F(x) as for /(.r). 



/•» . 



32. Whenever / /(*') dx is essentially convergent, we may at once put a = oo in the 



Jo 



preceding formulae. For, first, it may be easily proved that in this case, (though not in this case 

 only,) the limit of (53) when h vanishes is f(x) ; secondly, the limit of {53) is also the value of 

 (52) ; and, lastly, all the derivatives of f(x) have their integrals, (which are the preceding 

 derivatives,) essentially convergent, and therefore co may be put for a in the developements of the 

 derivatives in definite integrals. 



When f{x) tends to zero as its limit as x becomes infinite, and moreover after a finite value 

 of X does not change from decreasing to increasing nor from increasing to decreasing. 



r 



' f(x') sin ^x'dx 



