AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 191 
Art. 12 .— To prove that if U = 0 be the equation of the Locus of Uniplanar Nodes, 
A contains U 6 as a factor in general. 
(A.) Amongst the conditions satisfied at every point of the uniplanar node locus, 
will be found the following, see the ratios (48) :— 
[a, a] [/3, /3] — [a, /3] 3 = 0.(76). 
Now, by Art. 1 , Preliminary Theorem C, this means that the equations 
DfjDa = 0, D f/Db = 0 
are satisfied by two systems of values of the parameters which become equal when 
x = £ y = 77 , z = £, the coordinates of a point on the uniplanar node locus. 
[It must be remembered that the theorem has to be specially interpreted for the case 
in which Df/Da, Df/Db are both of the first degree in a, b, i.e., for the case in which 
/is of the second degree in a, b. This is done in Section IV.]. 
Now this is the case previously reserved in Art. 7, Art. 9 (the second case of 
exception), and Art. 11 (condition ( 68 )). 
As there are, in this case, two equal values of each of the parameters, a, b, it may 
be expected that there will be two (not necessarily equal) values of da/dx, 8 b/ 8 x. 
It will appear presently that in some cases 8 a/ 8 x, 8 b/ 8 x may become infinite, but 
this is not the case for the uniplanar node locus, in which the values of dafdx, 8 bf 8 x 
as given by (52), become indeterminate, because the conditions (48) are satisfied. 
Differentiating (52) again with regard to x, it follows that 
[x, x, af\ -j- 2 [x, rij, 
+ 2 o> «i. 6 i] 
06, 
8x 
+ 2 [«!, ®i. 6,] (!“') (fj + [«„ h, M (fr) + [<*i> “J + [“!• \ll - 0 • 
O. *, M + 2 [*, «„ 6,] + 2 [x, b lt &,] | + 0 „ «„ 6,] (|Y 
(77), 
+ 
2 K KM (f)If) + Pi ,K 6 .] (IT + [«.,+ Pi,Mai = 0 • (78). 
V 8x 
8x 2 
Multiplying (77) by [0,, bf\, (78) by [oq, b{], subtracting, putting x = £ y = 17 , 2 = £, 
and therefore cq = a, 6 , = (3, and using (76), it follows that 
