( 549 ) 



^ 3. In order now (o write down the equations which u and ?? 

 must satisfy, we can make use of {IV) and {VI), or we can use 

 the conditions of integrability. 



(i V,a) gives : 



bu bz — 2u dv du 2v / 2 du du 1 b^u 



drj ' 05 iQ{v—uy'd^ di] iQ\{v—uy d^ drj v — udidri 



(IV,b) gives : 



^u 0^ _ 2u dv du 2u f 2 bu du , 1 b^u 



4" T 



di^'dg iQ{v — uy'b^ bii iQ\{v—uy'di, d)] v—u ö^drj 



Out of {VJ) we find: 

 b /bz\ 2 u + w ÖM du 2y f 2 bu bu 1 dV< 





öëVÖ'?/ iQ{v~t<y'bri'b^, iQ{v -u)\{v—uybi br] ' ?;— mÖ^Ötj 



b /'bz\ 2 V -{- u bu bv 2u f ^ bv bv I b^v 



biiyb^J iQ{v-uy' bi^' bS, iQ{v — u)\{v—uy Ö5 bri v-ub^br] 



and 



d'2 Q fu v\ bz bz , b^'z __ 2 Ü 4- u bu bv 



Wn^2i\v'~ü)'bi'b7i ^'""^^ Wn~~ ÏQi^'üy'b^'bl; 



The equations given above show that all the conditions of the 

 problem can be satisfied in the only way by putting : 



2 bu bu 1 b'u , 2 bv bv 1 b^v 



= Oand— ; rT---T- + ; ^^t- = 0, 



{v — uyb^ bt] {v -u) bS,bri (v—nybB, by] (u — u)bS,bri 



which equations we write in the form : 



bu bu b^u \ 



bB, bt] dgot] 



.... {VII) 

 bv bv b^v „ 1 



0^ bri ^ ^bibii 1 



So tiie problem is entirely reduced to the integration of these two 

 sumultaneous difterential equations which are of order two and 

 non-linear. 



It is easy to deduce from ( VII), that the conditions 



b fbx\ _ b fbx\ b /by\ _ b /by 



are satisfied. 



We find namely always : 



b\v _ 2{uv — l) bu bv b'y _2{uv -\- I) bu bv 



b^i ~ ~~ Qi(v-iiy ' b^i' bÈ, ' ö|8^ ~ Q{v—uy ' ö7f d| ' 

 whilst 



