566 Mr. stokes, on THE CRITICAL VALUES OF 



become infinite for a = 0, whether h be finite or be zero. Threfore -— is suddenly decreased by 



da 



■TT as o increases through zero, as might have been easily proved from the expression for u by means 



of the known integral (71), even had we been unable to find the value of u in (72). The equation 



(74) gives, a being supposed positive, 



u =€£-"+ C'e". 



But u evidently does not increase indefinitely with a, and u = I = — when a = ; 



•^ •' ^ 1 + ,r- 2 



TT 7r 



whence C' = 0, C = — , m = - e'". Also, since the numerical value of u is unaltered when the 

 2 2 



sign of a is changed, we have m = - e" when a is negative. 



It may be observed that if the form of the integral u had been such that we could not have 

 inferred its value for a negative from its value for a positive, nor even known that ii is not infinite 

 for a = — OS , we might yet have found its value for a negative by means of the known continuity 



of 71 and discontinuity of --— when a vanishes. For it follows from (74) that m = 6^6" + C.,e~" for a 

 da 



negative; and knowing already that u = - e'° for a positive, we have 



-:=C, + C,, --=C, -C.-7r; 

 2 2 



TT TT 



whence Ci = — , Cj = 0, « = — £», for a negative. 



Of course the easiest way of verifying the result m = — e'" for a positive is to develope e"' for 

 a: positive in a definite integral of the form (59), by means of the formula (60). 



SECTION IV. 



JSxamjiles of the application of the formnlce priwed in the preceding Sections. 



43. Before proceeding with the consideration of particular examples, it will be convenient 

 to write down the formulfe which will have to be employed. Some of these formulas have been 

 proved, and others only alluded to, in the preceding Sections. 



In the following formulae, when series are considered, /(a?) is supposed to be a function of x 

 which, as well as each of its derivatives up to the {ft. - l)"" order inclusive, is continuous between 

 the limits a; = O and x = a, and which is expanded between those limits in a series either of sines 



or of cosines of — and its multiples. //„ denotes the coefficient of sin when the series is one 



a a 



of sines, B„ the coeflicient of cos when the series is one of cosines, -■/„'' or Bj^ the coefficient of 



a 



sin or cos in the expansion of the /«"' derivative. When integrals are considered, /(.r) 



a a 



