1886.] 



Solutions in the Calculus of Variations. 



±11 



taken in its unreduced form, and, by considerations founded on the 

 degree of continuity required in the variations of the dependent 

 variables, it is shown without difficulty that, when the range of integra- 

 tion is small, the sign of the second variation is the same as that of a 

 certain quadratic function. The limits of integration within which 

 this result is applicable are then determined by considerations de- 

 pending on the continuity of the integrals. 



The following is a brief sketch of the method of obtaining the 

 quadratic function on which the sign of the second variation depends. 

 Let the function to be made a maximum be — 



U= j| . . . fa H , . . «*« y v y p . ■ *| Jj, ^, - ■ • ■ ■ ■ ** 



(the form of y lt y 2 , . . . y n as functions of x v x 2 , . . . x m having already 

 been determined by making the first variation vanish). Taking the 

 second variation by Taylor's theorem, we may write — 



^U=ij j . . .. \^-^-hzhz'dx x dy 2 . . . dx m 



when z and z' typify any of the quantities y v y 2 , . . . y n , or their 

 differential coefficients, and 2 means that all such terms are to be 

 taken. Restricting ourselves to the case in which the limits are fixed, 

 we have By lt By 2 , . . . 8y n zero at the limits, and similarly any differ- 

 ential coefficient of these quantities must be zero at the limits, pro- 

 vided it appears among the limiting terms in the first variation. Now 

 if 6 be any function of x which vanishes when x=x Q , we have — 



4" 



J X 



de , 



— dx. 

 dx 



idO 



Hence, if x—x be a small quantity of order (3, 0/— is also a small 



dx 



quantity of the same or a smaller order, and therefore if we require 

 to obtain the sign of a quadratic function of and we may neglect 



all terms except those involving 6> 3 . Reasoning of this character is 

 applied to the variations in 6 3 TJ, and the important terms are thus 

 picked out. In this way a set of inequalities is obtained which 

 enable us to determine the important terms in S~U, and therefore its 

 sign. 



