550 Mr. stokes, on THE CRITICAL VALUES OF 



If we wish to find the direct expansion of /*'(«) in the case in which A„ or B, is given, we 

 have only to expand A„ or B„ in a series according to descending powers of v, regarding cos ny 

 or sin 7iy, as well as (- 1)% as constant coefficients, and then reject from the series obtained by the 

 immediate differentiation of (24) or (30) those terms which would render it divergent. This readily 

 follows as in Art. 15, from the consideration of the mode in which A'^ is obtained from A„ or B„. 



The equations ^35) and (36) contain as particular cases (26) and (32) respectively. It was con- 

 venient however to have the latter equations, on account of their utility, expressed in a form which 

 requires no transformation. 



18. If we transform A„ and B„ by integration by parts, we get 



2 »7ra 2o „„ . nira 2 a^ _,^ nira ^„_^ 



J^= — A'Qcos r^^Q, sin -—SQ., cos- + ..., (37), 



7nr a n'lr a n tt' a 



2 itwa 2a „ mra 2a- . nira , , 



»= ^Qsin- — ^Q,cos — + — -— ^Q.sin + ..., (38), 



where the law of the series is evident, if we only observe that two signs of the same kind are always 

 followed bv two of the opposite kind. The equations (37), (38) may be at once obtained from (35). 



(36). The former equations give the true expansions of A„ and B„ according to powers of — ; 



because when we stop after any number of integrations by parts the last integral with its proper 

 coefficient always vanishes compared with the coefficient of the preceding term. 



Hence A„ and B„ admit of expansion according to powers of - , if we regard cos ny or .sin ny 



as a constant coefficient in the expansion. Moreover quantities such as cos ny, sin ny will occur 



alternately in each expansion, the one kind going along with odd powers of — and the other along 



with even. If we suppose the value of A„ or J5„, as the case may be, given, and the expansion 

 performed, so that 



A„ = SF cos ny .- + SFi sin ny. -■ + SF.^ cos 7ly .— + ..., (39), 



n w w 



B„ = SG sin ny . —+ SGi cos ny .—„ + SG„ sin ny .—, + ..., (40), 



n n- n" 



and compare these expansions with (37) or (38), we shall get the several values of a, and the 

 corresponding values of Q, Q,, Q, ... We may thus, without being able to sum the series in 

 equation (24) or (30), find the values of x for which f{ic) itself or any one of its derivatives is 

 discontinuous, and likewise the quantity by which the function or derivative is suddenly increased. 

 This remark will apply to the extreme values and a of x if we continue to denote the sum of the 

 series by /(r) when so is outside of the limits and a. 



19. Having found the values of a, Q, Qi ..., we may if we please clear the series in (24) 

 or (30) of the terms which render /(.r) itself, or any one of it dervative.s, discontinuous. If we 

 wish the function which remains expressed by an infinite series and its first /i derivatives to be 

 continuous, we have only to subtract from A„ or jB„ the terms at the commencement of its expansion, 



ending with the term containing 1- , and from f(x) itself the sums of the series corresponding 



to the terms subtracted from A„ or B„. These sums will be obtained by transforming products of 

 sines and cosines into sums or differences, and then employing known formulae such as 



