AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 181 
In like manner (36) shows that (16), (18), (28) are equivalent to two independent 
equations only. 
Also (37) shows that (17), (18), (28) are equivalent to two independent equations 
only. 
Hence (28) and any two of the equations (16), (17), (18) are equivalent to two 
independent equations only. 
Similarly, by means of (38)-(40), it can be shown that (29), and any two of 
the equations (16), (17), (18) are equivalent to two independent equations only. 
Hence the five equations (16), (17), (18), (28), (29) are equivalent to two inde¬ 
pendent equations only in this case. 
Since (16), (28), (29) are not independent, it follows that (amongst other relations) 
and 
DJIB. W> 08]] 
D[ f, «, P ] 
(41), 
P[[fl, M. 08]] = D[[q], [«], Q8]] 
D [ v , a > 0 ] H [ £ . * , 0 ] 
Art. 6.— Investigation of the conditions which are satisfied at any point on the Locus 
of Uniplanar Nodes. 
In this case, the left-hand side of the equation (33) becomes a perfect square. 
Therefore 
- [y, a 
= K, a 
K’?] 
ly> y] 
K. y] 
[^] 
ly> Q 
& a- 
(43). 
Now, multiplying (16) by [y, £], (17) by [£ £], and subtracting 
(»«) {[f.»] h. f] - h.«] a. a + m {[f.« h, f] - h. P] [ft f]} = o . (44). 
Now, there is a uniplanar node on the surface (24), hence S/3 may be made to 
vanish. 
Therefore 
Similarly 
Therefore 
&«][?. a-fo.«][£a = o.(45). 
K ft fo. fl - lv, ft [£ fl = o. 
K a / b. a = [£ «] / «] = K ft/ lv> ft.( 4(; ). 
Now (43) and (46) show that (16) and (17) are equivalent to one independent 
