RECENT ADVANCES IN SCIENCE n 



that of minimising an integral f(x, Vi, . . . , y n , y\, . . . , y' n )dx, 



with respect to curves which join two fixed points and satisfy 

 a certain system of differential equations. For this problem 

 proofs of the necessity of the rule of Euler and Lagrange 

 and the conditions of Weierstrass and Legendre are obtainable 

 without the use of the second variation ; but the case of Jacobi's 

 condition is not so satisfactory. D. M. Smith [Trans. Amer. 

 Math. Soc. 1916, 17, 459-75) gives proofs of the necessity of 

 both the conditions of Legendre and Jacobi by means of the 

 second variation, but without the use of complicated trans- 

 formations. 



A. Dresden had obtained in 1908 formulae for the second 

 derivatives of the extremal integral arising in the problem of 



minimising or maximising the integral F(x,y, x' ,y')dt. In these 



formulas the derivatives were expressed in terms of particular 

 solutions of Weierstrass 's form of Jacobi's differential equation, 

 and were used to obtain further necessary conditions for a 

 minimum when one or both end-points were variable along a 

 curve or when curves with discontinuous slopes were admitted 

 as solutions of the problem. Dresden [ibid. 425-36) uses the 

 same general method in analogous problems involving integrals 

 of a more general type, and also in the cases in which the un- 

 known functions are further conditioned by differential and 

 algebraic equations. 



P. R. Rider (Bull. Amer. Math. Soc. 191 7, 23, 237-40) 

 gives the corner conditions and the forms of the Caratheodory 

 .^-function for discontinuous solutions in the calculus of varia- 

 tions for the form of the problem considered by Bliss (1907, 1908), 

 and for an analogous form of the space problem. We may refer 

 also to Rider's note in the Amer. Math. Monthly (191 7, 24, 1 34-6). 



On a theorem of H. M. Morse on the linear dependence of 

 many analytic functions of one variable there are three notes by 

 Morse, G. A. Pfeiffer, and G. M. Green in the Bull. Amer. Math. 

 Soc. for 1916 (23, 1 14-17, 1 1 7-1 8, and 118-22 respectively). 



T. H. Gronwall (Trans. Amer. Math. Soc. 1917, 18, 50-64), 

 completes Cousin's work on the expressibility of a one-valued 

 function of several complex variables as the quotient of two 

 functions of whole character by giving precision to the conditions 

 of two theorems stated by Cousin. 



