302 



LIPKA. 



\dx2 dxij \dv du du dv J V3.r3 dxij \dv du du dv J 



= 0, (u -^ V ~> w) 



To eliminate the /'s we proceed as follows. Solve (33') for the ratios 

 of the p's, thus 



(43) pi : P2 : Ih : Pi = 



rj/* rj/* '\^ 



Multiply the three equations of (42) by — , — , -^, respectively, and 



dto du dv 



add, remembering the expansions of the determinants (43) for the 



p's in terms of minors of the second order. We get 



/dFj _ dFA ^ (dT\ _ dF2\ _ 

 'V^i's dx2j \5.r2 dxij 



fdF\_dFs 

 \dx3 dXi 



= 0. 



A-f r)f T^ 



Similarly, multiply the three equations (42) by -^, — ^, — , respec- 



dw du dv 



,• 1 1 dfs djz dfs . , , , 5/4 dfi a/4 ,. , 



lively, by —,—,—, respectively, and by —,—,—, respectively, 

 dw du dv dw du dv 



and add. We thus get the four conditions 



^dF4 dF. 



(44) 



fdF, 



\dX3 



dFs 



' c).ri 

 fdFi 



dh 



dxo 



dFi 



dxs 



dFo 



+ P3 



- Ih 



+ V2 



dx, 

 fdFo dF,\ . 



\dX3 dX2 J 



a.r2 

 dFi 



M 

 dFi 



dx, 

 dF, 



dx'i 



dxi 

 dF\ 



dXi 

 dj\ 

 dXi 



dFi 



- p. 



+ lh 



/dFi dF; 



[d^ ~ 

 dFi 



dx: 



(9.r4 

 dFz 

 6.1-4 



= 



= 



/dFi dFA _ 



- Pi ^ — ^~ r ° 



Vo.r2 a.r4 / 



^^-^) + ;. 



dFi dFi 

 (9.C2 dxi 



= 



