AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 229 
Hence, at points on the biplanar node locus, 
a ~ c (uv - W 3 ) = 5 (U W - Yv) .(170). 
Again, multiplying the same equations by W. r , — u x , W, — u in order and adding, 
it follows that 
4(»-^=|(yw-n.) .(i7i). 
Again, multiplying (144), (155), (166), (167) by — XJ x , v x , — U, v in order and 
adding, it follows that 
a ( - WU,. + Yv x - UW, + vV x ) + (vw x + wv x - 2UU,) = X (vV - UW). 
Hence 
a^(VW-vV) = l(vw-U‘) . (172). 
Again, multiplying the same equations by V x , — W x , V, — W in order and adding, 
it follows that 
6 |(UW- 9 V) = t(UV-,„W). (173). 
Again, multiplying (143), (155), (165), (167) by U r , — W ( ., U, — W in order and 
adding, it follows that 
a (uU x - VW,. + U u x - WV X ) + (VU* - wW x + UV, - W w x ) = \ (uU - VW). 
Therefore 
« l (VW - «U) = l (UV - »W).(174). 
Again, multiplying the same equations by Y :c , — u x , Y, — u in order and adding, it 
follows that 
6|(VW-«U) = |(«»-V») .(175). 
Further comparing the three equations (168) or the three equations (169) with 
equations (l65)—(167), it is evident that it is possible in any one of the equations 
(170)-(175) to replace x by either y or 2 . 
It will be noticed that in the case of the biplanar node locus, the true values of the 
parameters may be found from any one of the ratios in (160). 
