AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 233 
Uy 
w 
VV y 
0 
u z 
w 2 
u 
u z 
w. 
u 
w z 
v z 
w 
+ 
W 
VV y 
Vy 
0 
+ 
w, 
v z 
W 
Y z 
v z 
Y 
Y, 
u~ 
Y 
Y y 
0 
Uy 
W, 
0 
u z 
w s 
u 
u z 
w z 
u 
W; 
v z 
w 
+ 
W 
W y 
v y 
0 
+ 
w, 
V, 
W 
v. 
u* 
Y 
Y z 
V z 
Y 
JUM 
- ClUy 
- bWy yW - 
- ClWy — bVy 0 
= ! x 
u, W, u 
W, v z W 
u W 0 
+ u y {Yv z - WU ; - a (WW Z - uv z )} 
+ W, |WV ; - 2YW. + uU z - b (WW : - uv z ) - a (uW z - Wu z )} 
+ Vy {Yu, - uY z - b (uW z - W u s )} 
= - fa ( uv ~ W3 ) 
+ u y { v z (ciu + Y)-W (aW z + U*)} 
+ W y (W (au z + Y z ) — W z (au + bW + 2Y) + u (bv, -f- U.)} 
+ v y {u z (bW +Y)-u (bW z + V,)} 
= ~ P l %( uv ~ W 2 ) 
~\~Uy (— Z/W 2 ) + W y ( 2VUW) + Vy (~ VU~) 
= - jxu ^ (uv - W 2 ) - m ~ (uv - W 2 ). 
Hence the coefficient of y. obtained from this by changing z into x, and y into z ; 
and, therefore, v into X, and /r into v, is 
vu ^ (uv — W 2 ) — \u (uv — W~). 
dz 
And the coefficient of v, obtained by changing, in the coefficient of X, y into x, and 
2 into y , and therefore, /r into X, and v into /r, is 
- ku^(uv - W 2 ) - fiu ~ (uv - W~). 
From these the equation (179) follows. 
MDCCCXCTI.-A. 
2 H 
