AND LINES JN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 249 
The last determinant by (178) is equal to 
- 6X z u ~ {uv -W 2 ). 
Hence 0 3 A/0a ; 3 = 0. 
Next take the value of 0 3 A/ 0 x 2 02 / from (151). 
The first three terms vanish by (158). 
The next three terms by (177) are equal to 
+ «-) l (- - W=) 
= 2 \*U ^ {uv ~ W 2 ). 
The next three terms by (177) are equal to 
0 
2 {a?u xy 2ctb~W xy -f- b~v xy -f- 2 ccV xy -f- 2bXJ xy -f- w xy ) ^ {uv — "W ~) 
= 4:\[xu {uv — W 2 ). 
The next three terms may be obtained from (179) by changing z into x, and, 
therefore, v into A. They are therefore equal to 
— 4c\[jlu ■— {uv — W 2 ) — 2\ z u {uv — W 2 ). 
Hence 0 3 A jdx^by = 0 . 
Next take 0 3 A/0cc0y0z from (152). 
The first three determinants vanish by (158). 
The next six are by (179) 
0 
LLVU 
The next six are by (177) 
= {<^u xy + 2abW xy + b 2 v xy + 2 aV xy -f 2bV xy + w xy ) I {uv — W 2 ) 
= 2A fiu y {uv — W 2 ). 
Hence, the next six are 
= 2 ^ vu 51 ( uv ~ w )> 
dx 
2 K 
MDCCCXCII.—A. 
