AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 261 
The last determinant is by (178) 
= - 6\*u^(uv - W 2 ) = 0. 
Therefore 3 3 A/d,x 3 = 0. 
Next take 3 3 A/3 x^dy as given in (151). 
The first three terms vanish by (158). 
The next three 
= k„|(oto - U-) + 2W«|(UV-W#) + V 2 ) 
+ 2V«| (WU - Vv) + 2V.„| (WV - U«) + (uv - W 2 ) 
= 0. 
The next three 
= 2 u„ £ (vw - U 2 ) + i W, l (UV - W w) + 2», l (uw - V 2 ) 
+ 4V„ | (WU - Vv) + 4U,| (WV - U«) + 2 w„ l (uv - W 2 ) 
= 0. 
The next three may be calculated by means of (179) by putting z = x, and there¬ 
fore v = X. 
Hence they are equal to 
- 4A ijlu ~ (uv - W 2 ) - 2 X z u ~ (uv - W 2 ) = 0. 
Hence 0 3 A jdx^dy = 0. 
Next take 3 3 A /dxdydz as given in (152). 
The first three terms vanish by (158). 
The next six terms by (179) 
= — 2/xz vu (uv — W 2 ) — 2v\u (uv — W 2 ) — 2 \/jlu v- (uv — W 2 ) 
= 0. 
The next six terms 
= «, 3 (vw - U 2 ) + 2W,| (UV - Ww) + («w - V 2 ) 
+ 2V, ~ (WU - Vv) + 2U,| (WV - U u) + JO, | (uv - W 2 ) 
= 0. 
