[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 123 
On developing these oT gage we find for ba eer and third: 
rs) 06 
= (Gs) - (es Pi +ontbe) À PRE 45 (pit ° Phares = —2a a'P3 ae == 
Gt ee — 2b’ ps - 
Ê ae Gr + >) ant Lx + 




Sue = © —16aa'b—2 —- - Ge [a (ab) + 
/ / / a 9° 2b pee 
45 < (bb’) —8b(ab’ +a'b) —2 —— ~ 
00 
= - + gi + *) a aaa ae Fr aq) aa 
(aa += an 2e) à dv 
0 00 
~ du \ ou?ov J’ 
ae similarly 
90 ary ar. , \ 80 
ola (oa. nn *) aa 5 Cr: Re rs) aie 
(re + Tor so 
‘Ov 
_ fas 20 
- (Sit sats a5 sa + (ee +32 —2a ass) + 
CE — — 2054) - 
Lo (90 
~ du \ dv3 
The second - eee (3) aS on developing: 
0 f 00 00 
ov le e tant — dudv a (ant F Bes 0) 35 ou Le 
08 
(rts — — 20 q mF 
050 ~ 
= GE aac ani À) pr Oudv aE (na+ 5, uy 720" re) a + 
(ra 2br DE 
Thus in order that the second of equations (3) may be an identity, 
the following integrability conditions must be satisfied identically: 
Ogi ae or: 
au =F gar == G2 = ay PUNT 


