544 Mr. stokes, on THE CRITICAL VALUES OF 



of which H is the ordinate be the discontinuous curve AB, CD, EFG. Take Gb equal to BC, 



and on the positive or negative side of the axis of w according as the ordinate decreases or 

 increases as x increases through OM, and from measure an equal length Oc on the opposite 

 side of the axis. Take Gd, Oe, each equal to DE, and draw the right lines AG, Ob'b, cc'G, 

 Od'd, ee'G. Then it will be easily seen that if ^o' -^i' ^2 ^^ '^^ values of ^ corresponding to 

 the critical values of x, x = 0, x = OM, x = ON, respectively, X(, will be represented by the right 

 line AG; X^ by the discontinuous right line Ob', c'G; and X.^ by the discontinuous right line 

 Od', e'G. Take MP equal to the sum of the ordinates of the points in which the right lines lying 

 between OA and c B cut the latter line; MQ equal to the sum of the ordinates of the points in 

 which the right lines lying between c B and d'E cut the former, and so on, the ordinates being 

 taken with their proper signs. Let P, Q, R, S be the points thus found, and join AP, QR, SG. 

 Then SX will be represented by the discontinuous right line AP, QR, SG. Let the ordinates of 

 the discontinuous curve be diminished by those of the discontinuous right line last mentioned, and 

 let the dotted curve be the result. Then 3 - SX will be represented by the continuous, dotted 

 curve. It will be observed that the two portions of the dotted curve which meet in each of the 

 ordinates MB, NE may meet at a finite angle. If there should be a point in one of the con- 

 tinuous portions, such as AB, of the discontinuous curve where two tangents meet at a finite angle, 

 there will of course be a corresponding point in the dotted curve. 



If we choose to take account of the conjugate points of the curve of which SX is the ordinate, 

 it will be observed that they are situated at O, and midway between P and Q, and between R and S. 



9. There are a great many series, similar to (3), in which f{x) may be expanded within 

 certain limits of x. I shall consider one other, which as well as (3) is of great use, observing that 

 almost exactly the same methods and the same reasoning will apply in other cases. 



The limit of the sum of the series 



- / f(x) dx + - Sfi-" / /(>t') cos dx' 



n J^, a J,j a 



(21)' 



