4-10 Mr. I. Todhunter on the Calculus of Variations. 



= e" 2 « 2 1 , £ arc tan ( — t=£= • — c ± dp. 



or, putting p 2 = #, 



{Erf (*)} 4 



= e - 2 a 2 f 2 / _iT 2,a? arc tan^f±?A /*El)dx. . (9) 



If we differentiate (9) with respect, to a, we obtain an expres- 

 sion for (Erf a) 3 as a definite integral, viz. : — 



4 {ErfW} 3= 2a j i -^aretang^y^)^ (10) 



Of the three formulae (7), (9), (10), the first may open a series 

 of interesting integrals ; the other two are perhaps too complex 

 to be any thing but a matter of curiosity. 



Cambridge, October 31, 1871. 



LVI. On a Problem in the Calculus of Variations, 

 By Isaac Todhunter, F.R.S. 



To theEditors of the Philosophical Magazine and Journal, 

 Gentlemen, 



IN the Philosophical Magazine for July Professor Challis has 

 discussed three problems in the Calculus of Variations. 

 He states, in connexion with the first problem, that a certain 

 conclusion obtained by Legendre and Stegmann, and tacitly ac- 

 cepted by myself, is erroneous. 



There is, however, no error. Professor Challis does not un- 

 derstand the problem in the same sense as Legendre and Steg- 

 mann. I have enunciated the problem thus: — required to con- 

 nect two fixed points by a curve of given length so that the area 

 bounded by the curve, the ordinates of the fixed points^ and the 

 axis of abscissae shall be a maximum. It is intended that the 

 curve should be confined between the indefinite straight lines which 

 coincide in position with the extreme ordinates. The enuncia- 

 tion involves this condition ; for otherwise nothing would be 

 gained by introducing the ordinates of the fixed points. And 

 the investigation given would show, if there were any doubt, that 

 this is the precise meaning intended. 



It is in fact this condition which constitutes the chief interest 

 of the problem ; and there can be no doubt that the solution of 

 Legendre and Stegmann is correct. Stegmanu's investigation 



