By Marston Morse

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**Sample text**

I Here D(A)/D(y) I 0 as we have stated in. 6). 16). re independent of the choice of the regular representation [ r( v)] of class C" ot the indicatrix f(x , r) 0 =l neighboring the initial direction (u). ' . another regula1· repreaente,tion of the indioatrix in the neighborhood of (u) 59. is used in terms of parameters iation v . i = mi(v) (i (v), = l,. to the parameters (v) will stand in a re- i the parameters·(v), where the functions m . are of class_ C" and have a non-vanishing jacobian at (v) = (o).

Eters 0 the integral along the ex- tremala issuing from P is a function (s J(s, v) = )0 An aro b o~ F[x(s, v), • x(s, v)]ds. Zc hns the form vh= vh{t) {h = l, ••• ,n). ::i u(x) ]xi •. ~ To show that Z is a topological {n-1)-sphere one uses the functions Ai c ot Theorem. 1 to represent Z0 .

He extremal at which s = f(x , y). ·- * That 0 0 is, 1-1 and continuous both ways. 39. 9) for t, with (y) = (r 0 ), includes t = l. 13) = x defining Ai in this way. i Observe that Ai is of class C" in (y) for (y) I (o). 14)a We set (y) OA,i ---:- (x , o) OyJ o = t(u) = 5. 12). 15) provided (r) is sufficiently near a unit contravariant vector (u), For, the condition t < e(x 0 ) implies the condition . t f(x , r) < e(x ) 0 0 if (r) is sufficiently near a unit vector. 16) oxi CA. i t---:- (x, tr)=---:- (t, x, r) Oy-J 0 orJ 0 Reoall that r) = xi0 J x 0 ' r) = O < Cr> I