180 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
[£ f] (X ~ f )* + [ V , r,} (Y - yjf + [{, Q (Z - £)* 
+ 2[ v ,t](Y- v )(Z-£) + 2[lJ](Z-0(X-f) + 2[£, v ](X-£)(Y- v )=.0 . (33) 
This breaks up into factors, linear with regard to X — Y — rj, Z — £. 
Therefore 
DIHIMjrn _ 0 
viz, v, n . 
(34). 
Now if in (16)—(18), S/3 be put equal to zero (which is possible, since there is by 
hypothesis a biplanar node on the surface (24)), the values of S£/Sa, Srj/Sa, Si,/Sot must 
be finite. 
But the denominators of the values of these expressions vanish by (34). Hence 
their numerators also vanish. 
Therefore 
P[[BM»H] = P LIB M. [fl ] _ 0 
h[?.^. n P[ £ , v , * ] 
p[[a ra. r«n = p[[a. w, rai = 0 
p [£>*?» £ ] p[^.?>«] 
p[w. M i = p [ca w. m i = o 
p[ f, v, r ] p[, r, «i 
(36), 
(37). 
And similarly it can be shown that the following equations obtained by changing 
into /3 in the above also hold good :— 
p l iB m. [j3u = pitaM [fl] = f) 
P[ £, V , i ] P[ £ , v , /3 1 
p[ [fl, ca pa]} = p[[g. w, mi = 0 
p[ f m d[ *, r, i3] 
p[[iy].m. [ 0 ] = pnaM_m ] = 0 
P[ f , *?, M D[ 17 , ?, £ ] 
From these it follows that (1G)-(18) are equivalent to two 
only in this case. 
Now consider equations (16), (17), (28). 
The equation (35) makes the determinant formed from the coefficients of S£, 8 rj, Si, 
in them vanish. Hence it bears to them the same relation that (34) bears to 
(16), (17), (18). 
Therefore (16), (17), (28) are equivalent to two independent equations only. 
.(38), 
.(39), 
.(«). 
independent equations 
