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By Dorothy L. Bernstein

The description for this booklet, lifestyles Theorems in Partial Differential Equations. (AM-23), may be forthcoming.

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362-373) but these are not existence theorems and so are not included here. For n+l independent variables, the statement and solution of problems G, I, C are where z direct gener~lizations of the problems for 2 independent variables, = g(y) replaced by x is replaced by z = g(y1 1 ... 5) x where II~ J ~0 ... ~~~ 011 If the determinant j is of rank II. g~tol r 0, one can solve n of the equations (13. 5) for µ1, •.. nsformed to problem IN, and the theorems of section 11 applied. • ,qn of µo, ... ,µn such that 0 0 0 0 0 0 0 F [ x ,y1, ...

Let zo = g(yo) and qo = g'(y 0 ). 33 00 (b00 ) Let f(x,y,z,q) be a function of class A in S0 : I x-xol O, < IY-Yol Then there exists a unique function o, < ~(x,y) lq-qol < O. 5) Proof: IY-Yol < 01, For all (x,y) ER0 , z = < 02 of (xo,Yo). ~(x,y) ( 7. 5) ~ = f(x,y,z, ~) ~(xo,y) : g(y). See Goursat 1 , Horn 1 , is a solution of: pp. 163-165. ,µ,v=o where this series converges absolutely in S0 . We wish to determine, first formally, the coefficients of (10. 8) g(y) where this converges absolutely in T0 .

2)· Proof: ~ F(x(µ),y(µ),z(µ),p(µ),q(µ)) = Fx · x' + FY • y' + Fz z I + Fp • p I + Fq · q' = FxFp + FyFq + Fz(pFP +qFq) + Fp(-Fx-PFz) + Fq(-Fy-qFz) = 0. F(x(µ),y(µ),z(µ),p(µ),q(µ)) = C, a constant, for all µeM. 4) and is therefore an integral strip. 2) which is of class A2 in a region R. 11s this element for µ = 0 and which lies on z = ¢(x, y); this is an integral strip. 2). solution x = xo(µ), y e~ists a yo(µ) of the equations: x' Fp(x,y,¢(x,y),¢x(x,y),¢y(x,y)) y' = Fq(x,y,¢(x,y),¢x(x,y),q\(x,y)) which forµ= 0 passes through (xo,yo>.

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