AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
269 
for, if so, [£ f] X + [£ rj ] Y + [£ Q Z would be a multiple of [£77] X+ [77, 77 ] Y + [77, £]Z, 
and the biplanes would coincide, and there would be a uniplanar node. 
Consequently, in this case the equations can be reduced as follows :— 
The lowest terms in (183) to be of the second degree. 
The lowest terms in (184) to be of the first degree. 
The lowest terms in (185) by means of (184) to be of the second degree. 
Hence the degrees are respectively 2, 1, 2. 
Hence there are four intersections. 
Art. 29.— To prove that the Surfaces represented by the three fundamental equations 
intersect in six 'points on the Uniplanar Node Locus. 
In this case 
[£ a X 3 + [77,77] Y 3 + K, Q Z 3 + 2 [77, Q YZ + 2 B, |] ZX + 2 [£ 77] XY 
is a perfect square, and is proportional to the square of [£, (*] X + [£, 77] Y + [£ £] Z ; 
and this by means of the ratios (48) is proportional to a] X 4- Tti, a] Y 4- [L oil Z 
and also to [£ 78] X + [77, 78] Y + [£, 78] Z. 
Hence the lowest terms in X, Y, Z may be reduced as follows :— 
The lowest terms in (183) to be of the third degree; the lowest terms in (184) to 
be of the first degree; and the lowest terms in (185) by means of (184) to be of the 
second degree. 
Hence the degrees are 3, 1, 2 respectively. 
Hence there are six intersections. 
Section YI. (Art. 30).— Exceptional Cases. 
Art. 30. 
It remains to notice the exceptional cases in which the locus of ultimate intersec¬ 
tions is not a surface. 
A 11 example is given of each, but the theory is not developed. 
The general case which has been considered in this paper is that in which the 
fundamental equations are satisfied by values of the coordinates which are functions 
of both parameters. 
The exceptional cases are :— 
(I.) When the fundamental equations are satisfied by values of the coordinates 
which are functions of one parameter. 
(II.) When the fundamental equations are satisfied by values of the coordinates 
which are functions of neither parameter, i.e., are independent of the parameters. 
