Mr. John Walsh on the Calculus of Variations. 831 



not still for the determination of the maximum, but the ori- 

 ginal equation, x 



adx — 2xdx = alx — 1xlx-=. 0. 



I shall take now what is called the differential equation of 

 the curve of quickest descent. Finding the variation, and in- 

 tegrating, then 



*+/! 



■j isdu dsiu j 



The terms remaining under the integral sign destroying one 

 another, there is still for the determination of the minimum, only, 

 the original equation, i s ds 



= 0. 



Thus, it is first of all demonstrated, by the fifth proposition 

 of the second book of Euclid, that the calculus of variations 

 is no calculus at all. Its nonentity is self-evident; for it con- 

 sists in ascending from the second term of the development 

 of a binomial to the third, and then redescending again, leav- 

 ing us at the point at which we set out; doing and then un- 

 doing to no purpose. The Institute of France was too hasty 

 in the reports which it gave of every paper, in fact, which I 

 addressed to it. 



" M. Walsh croit que cette demonstration est la seule ri- 

 goureuse qui ait ete donnee jusqu' a present pour le cas dont il 

 s'agit.^ Cependant il suffit d'avoir lu les ouvrages d'Eulerou 

 des geometres qui ont ecrit apres lui sur cet objet, pour etre 

 bien convaincu que la formule du binome est depuis long- 

 temps etablie en toute rigueur."— Extrait du proces-vcrbal de 

 la Seance du Lundi, 24 Decembre 1821. 



" L'auteur s'est propose d'etablir un theoreme qu'on peut 

 exprimer comme il suit : Chaque terme du developpement de la 

 n me puissance du binome surpasse en valeur numerique la 

 somme des termes suivants. — Ce theoreme suppose evidemment 

 que le second terme du binome a une valeur numerique in- 

 feneure a celle du premier." — Lundi, 24 Fevrier 1823. 



With respect to the development of a binomial when the 

 exponent is a negative whole number, let 



(x + h) = = — _. 



(*+A) n x n +z 



developing the right hand member by division, and arranging 

 according to //, I get, 



The preceding is the demonstration to which the first of the 

 * l 2 preceding 



