Über dir unmerisclic Integration von. Différent ialglcichioigen. 



26 



Nr. IV: 



(111,111.2-3)' 



A'it = f (i,x,y,u, V) M , 



Vu ^ f(l +^A(,j; + j:A'.r,y + = A'ij,u+ = A'm,v + = A' v)M, 



A"'u = / ( t -t- ^ At, a; + ^ A'j, y + ^ ^'y, u -\-^^"u, î) + = A" y) A«, 



A'v = g(t,x,y,u,'i)) M, 



A'r = (7(« +|A(,.r + = A'x-,?y + |A'y, ti + gA' w,ü + ^A' 1)) A«, 



A'" r = g ( ( + ^ A f , a: + ^ A'x-, y + = A'i/ , w + = A"u ,v + "^ A" v )M, 



Ax =[m + ^^(2A'm + A"i( + A"'n)jAt, Au = ^[2 A'u + 3 ( A"m 4 A"'m)|, 

 Ay =[« +^(2A'y + A"(i + A"'7;)]A(, Au = ^[2 A'î; + 3 (A"« + A'" î;)|. 



A'a; = (m +gA'M) At, 



A'y = (i;+^A'i;)A<, 



Nr. V: 



(IV, lV.3-4)*) 



]Sr. IV: 



(V, IV. 4 5) 



A'w = f (t,x,y,u,v) M, 



^"u =/(« + 5A«,:r+^A'.<;,y+^A'y,M + ^A' m.v + ^A' y)Af, 



A"'n =/(/, + iAi,.r+^A'a:,y+^A'y,M + ^A"'M, îJ + ^A" v)M, 



\""u = f{t+ M,x+ à"x,y-r A"y.u+ A"'u,v+ A'" y) A«, 



A'v = g{l,x,y,u,v)M, 



à"v =g{t + ^M,x + !^à' x,y +^à' y,u + lA' u,v + r,à' v)At, 



A"'v = y (r + i At, 3- + ^ A' a; , y + J A' y, w + ^ A" M, y + i A" ») A(, 



A'a; =[M+iA' MjAf, 



A"a: = [M + ^A"'M]A«, 



A'y =b+jA' v]M, 



^"y = \v + '-^"'v]^t, 



A""v=g{t+ At,x+ A"x,y+ A"y,u+ A"'u,v+ A"'v)At, 



A;r = [u +l(à'u + A"m + A"'m)] A{, Au = J[A'm + 2 A"u + 2 A"'u + A""u], 



Ay = \v + l(A' V + A" V + A'" v)\At, Av = ^[A'u + 2 A" v + 2 A"'t; + A""v]. 



Ausser den Formeln in Nr. V werden, zwecks einer schärferen Bestimmung von 

 Aa;, Ay, folgende Ausdrücke berechnet: 



ô,x =l[u + j^{5A'u + A""m)] A«, â^u = ^^[B A'u + 8 A" a + 2 A"'w + A""u], 



Ö,y =3|^, +_L(5 A'v + A""i;)| At, ô,v=l^[bA'v + 8A"i. + 2A"'v + A"" v], 



A^'w = /(f +jAt,a; + ô^x,y + ôiy,u+ ô^u.,v ->r à^v) At, 



3 ' 



A^'v =Sf(t + jAf,x + d,,x,y + ô^y,u + ô^u,v + ô^v)Ai, 



Ax = [u + ^(13 A'u + 21 A"m + 21 A'" w + 6 A"" u - 16 A^'m)J Af , 

 Ay = [w + yç, (13 A' « + 21 A" î; + 21 A'" t; + 6 A"" y - 16 A^' v)\ Al. 

 Für Aw, Av gelten die in Nr. V berechneten Werte. 



N:o 13. 



