Equations of the Second Order. 31 



and integrating again, 



(* + c) 2 +(y + c') 2 = « 2 . 



Thus in this example Ada: and Apdx are both exact differen- 

 tials, and the integrations give the same result, viz. the equa- 

 tion of a circle the radius of which is a, and the coordinates of 

 the centre — c and — J. These three arbitrary quantities are 

 always determinable from the given conditions, provided the 

 given length be less than the straight line joining the two points. 

 Hence in all cases the required line is an arc of a circle termi- 

 nating at the given points. 



Mr. Todhunter states (art. 366) that this problem is discussed 

 by Legendre, and also by Stegmann in a work published no 

 longer ago than 1854, and that these two geometers arrive at 

 the same result, viz. that the required line is in some cases a 

 circular arc, and in other cases is composed of a circular arc and 

 one or two straight lines. The foregoing solution proves that 

 this cou elusion is erroneous. Mr. Todhunter has failed to 

 notice that the error is due to not recognizing the principle of 

 integrating with respect to consecutive points of a curve. 



Problem II. Required the minimum surface generated by the 

 revolution of a line joining two given points in a plane passing 

 through the axis of revolution. 



According to the principles already laid down, the surface is 

 fZiryds, ovf2ir\/ 1 +p 2 dx, whatever be the form of the line, 

 and wherever in the plane the given points be situated, if the 

 integration be supposed to be taken along consecutive points of 

 the line. Hence, by the Calculus of Variations, 



$fy4TTpd%==6, 



and V=?/Vl -\-p 2 . Consequently 



Therefore 



N- ( d? )= X - yq =-A- 



Vi+p 2 (i+p*)$ 



and 



1 x/l+Z (I +/>*)* 



Hence, by integrating, 



y = c\ / l+p 2 , 



which, as is known, is the differential equation of a catenary 

 having its directrix coincident with the axis of revolution. It 



