240 
MATHEMATICS: A. DRESDEN 
( 
where i^j, k = 1, . . . , n; r = 1, 2; r -\- s = 3; and dj 
/ = 1 when j = k 
\ = 0 when j k. 
4. For the integral (3), there is not in the literature a theory of the 
second variation and of Jacobi's equation analogous to Weierstrass's 
work for the integral (1). As an aid towards securing such a theory, the 
author estabHshed conditions under which it is possible to reduce a set 
of n Knearly independent linear differential forms in # + 1 functions to 
a set of n such forms of the same order in n linear combinations of these 
n -\- \ functions.^ This reduction is now used to carry through a re- 
duction of the second variation and of Jacobi's differential equations 
for the integral (3), analogous to Weierstrass's reduction for the integral 
(1). This reduction is dependent upon the non- vanishing of one of the 
functions y'i. In every regular problem one is assured that there is at 
every point of the interval (/1/2) a function ji, whose derivative does not 
vanish at the point, but the function may be a different one at different 
points. By the use of the Heine-Borel theorem, we can then, in virtue 
of the continuity conditions on the functions ji, divide the interval 
{tii^ in a finite number of intervals in each of which the reduction is 
possible. We consider now the system of reduced Jacobi equations 
which exists in the neighborhood of the point t — to and express the 
second derivatives of our extremal-integral in terms of sets of fimda- 
mental solutions of these equations. These solutions are known to exist 
throughout the entire interval (^1/2), even though the equations which 
they satisfy may not be valid beyond a sub-interval. In this manner 
a set of formulae is secured, which may be immediately applied for find- 
ing analytical conditions for a minimum of the integral (3), when one 
of both endpoints are allowed to vary in manifolds of 1 , 2 . . . , w — 1 
dimensions. 
It is intended further to develop similar formulas for the problems of 
minimizing integrals (2) and (3) if the unknown functions are further 
conditioned so as to satisfy a system of differential equations, or a sys- 
tem consisting of differential and of algebraic equations. 
1 A. Dresden, Trans. Amer. Math. Soc, 9, 467 (1908). 
^ See Bolza, Vorlesungen iiber Variationsrechnung, p. 233 and p. 312. 
3 Ibid., p. 316-318 and p. 328-330. These conditions had been known before. 
nbid., p. 372-389. 
