204 
PROFESSOR M. J. M. HILL OX THE LOCHS OF SINGULAR POINTS 
By (102), (103), and (104) it follows that 
l[a, a] ( 8 a ) 3 + [a, /3] (Sa) (8/3) + \ [/3, /3] (8/3 f 
+ terms of the third and higher orders = 0 
Hence ultimately 
[a, a] (Sa ) 3 + 2 [a, /3] ( 8 a) (S/3) + [(3, f3] (S/3 ) 3 = 0 . 
This determines the two values of the ratio 8/3/Sa. 
Then to determine 8 ^, Sy, S£ there are the following equations obtained from (98), 
(103), (104), by retaining only the principal terms. 
[£] ( S 0 + M (fy) + [£] ( S £) — 0 .(105), 
[a, £] (S£) + [a, 77 ] ( 877 ) + [a, £] (S£) + [a, a] (Sa) -f [a, /3] (8/3) = 0 . (106), 
[A £] (S£) + 1/3, 17 ] (S 77 ) + [/3, £] (SC) + [/3, a] (Sa) + [/3, £] (8/3) = 0 . (107). 
Hence the ratios S£ : S 77 : SC can be determined. 
Hence the directions of the tangents to each of the branches of the curve of inter¬ 
section of the envelope and the surface (9) can be determined. 
If, now, the condition (76) hold, the two values of S/3/Sa become equal, and, there¬ 
fore, the two tangents at the double point of the curve of intersection coincide, and 
therefore, the contact is stationary. 
Further, because in this.case the values of 8 / 3 /Sa both become equal to 
— [«, a ] / [a, P\ = — [«. £] / [A £]> 
therefore (106) and (107) become 
[a, £] ( 8 |) + [a, 77 ] ( 877 ) + [a, £] (S£) = 0 , 
[ft £] ( s O + [ft i\ (%v) + [ft Q ft 7 ?) — °- 
From these two equations and (105) it follows that the coinciding tangents at the 
double point of the curve of intersection lie in the tangent planes to the surfaces 
D//Da = 0, D/7 D/3 = 0,/= 0. 
(B.) In this case 
A = B f (x, y, z, cq, b 1 )f(x, y, Z, %, b 2 ) 
= W 3 .. (108). 
Hence A = 0, dA/Sx = 0 ; for f x ~ 0,/ 3 — 0 at every point on the envelope locus. 
Hence A contains E 3 as a factor. 
