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LVI. On the Calculus of Variations. By John Walsh, Esq. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



DERHAPS you will be so good as to insert the following 

 ■■■ paper in the next Number of your very valuable Journal, 

 if not too much preoccupied. 



I have the honour to be, gentlemen, 



Your much obliged servant, 

 Sept. 30, 1824. John Walsh. 



Let AB be any straight line bisected in C, and let E be 

 any other point in AB, then 



(AB-AC + CE)(AC ± CE) = AB x AC-AC" 2 ± 



(AB-2AC)CE-CE 2 . 



or, as AB is equal to 2AC, 



(AB-AC + CE)(AC±CE)=ABxAC-AC 2 -CE 2 . 



The preceding is the demonstration of prop. v. book 2, of 

 the Elements of Euclid. We see by it, as CE 2 is negative, 

 whether CE itself is positive or negative, that the maximum 

 rectangle under the two segments of any given straight line 

 takes place when the line is bisected. The condition of maxi- 

 mum requires, therefore, that the coefficient of CE the arbi- 

 trary magnitude added to and taken from AC, should be no- 

 thing, and that CE 2 should be negative. 



Clothing the preceding equation in the notation of the 

 differential calculus, there is given, in the case of maximum, 



adx — 2xdx = 

 in which a = AB, x = AC, dx = CE, dx, being considered 

 indefinitely small. Clothing this last in the notation of the 

 calculus of variations, there is given, in the case of maximum, 



fl(adx — 2xdx) = 0. 



f \adlx — f lxdlx— 2lxdx} = 0. 



Integrating by parts, the integral is, 



alx — 2xlx + /* ^2,%xdx — c llxdx}. 



It is seen, that the terms remaining under the integral sign 

 destroy one another. We are left, therefore, by the calculus 

 of variations, at the point at which we set out, and we have 



not 



