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1. First existence theorem. We continue the notation of preceding papers with the trifling change that z will denote (zl, , z ) instead of (zO, , zx) as heretofore, and a similar change for z'. The class of all rectifiable curves joining two fixed points zx, z2 (not necessarily distinct) will be denoted by K. The functions F(z, z') and G(z, z') are defined for all points z = (zl, , Zq) in a closed point set S and all z'. They are positively homogeneous of degree 1 in z', are continuous for all z in S and all z', and possess partial derivatives of first and second orders continuous except at z' = 0. We write

Some existence theorems in the calculus of variations. IV. Isoperimetric problems in non-parametric form