Principles of the Calculus of Variations. 49 



we have the three equations 



I take occasion here to remark that, in my communication to 

 the March Number (p. 220), I have not correctly indicated the 

 signification of the factor y. Since, after putting C = 0, the 

 equation is satisfied by y = 0, it is requisite to ascertain the 

 meaning of this result. Considered by itself, it means that a 

 straight line coincident with the axis of x is an answer in part 

 to the proposed question. Now as the equation y = is not in- 

 consistent with the equation y = c, I have no reason to deny that 

 that solution of the problem which is maintained by Mr. Tod- 

 hunter in the June Number (p. 426) is one which the analysis 

 includes. As, however, it gives a broken line — namely, a semi- 

 circle having its diameter coincident with the axis, and its extre- 

 mities connected with the points P and Q by straight lines — it 

 does not supersede the inquiry on which I am engaged, which 

 is, to find the continuous line that fulfils the condition of a maxi- 

 mum. Also, as the solution I refer to has sufficiently accounted 

 for the equation y = 0, we may, in prosecuting the other inquiry, 

 omit this equation, and use instead of it y — c, joining with this 

 equation the integrals of the two foregoing differential equations. 



These equations give by integration 



?/+ (*+ c') 2 =X 2 , i/+ (a; + c")*=X 2 ; 



so that we have at disposal three equations containing the three 

 arbitrary constants c, c 1 , c" for drawing a line from P to Q. It 

 will, I suppose, be admitted that, consistently with the principles 

 of the calculus of variations, the line may be composite in its 

 character : what we have to ascertain is, whether the analysis 

 indicates a continuous line — that is, one of which the component 

 parts have a common tangent at each point of junction. Since 

 two of the equations give equal circles having their centres at 

 arbitrary positions on the axis of x, the line may evidently be 

 drawn by making one of the circles pass through P, and the 

 other, with its concavity turned in the opposite direction, through 

 Q, and connecting them with the straight line parallel to the 

 axis of x given by the third equation. But this would be no 

 answer to the question unless the analysis indicated the mode of 

 junction, and the positions of the points of junction, of the three 

 lines. To ascertain whether it does so, recourse must be had to 

 the part of the variation freed from the sign of integration. If 

 the integral be taken from y = y to y = y v this part, by being 

 Phil. Mag, S. 4. Vol. 32. No. 213. July 1866. E 



