Dc 
XLVITI. 
ON HAMILTON’S SINGULAR POINTS AND PLANES ON 
FRESNEL’S WAVE SURFACE. By PROFESSOR WILLIAM 
BOOTH, M.A., Hoogly College, Bengal. 
[Read Frpruary 19; Received for Publication Frpruary 21; 
Published June 15, 1896.] 
[COMMUNICATED BY PROFESSOR T. PRESTON, M.A., F.R.U.I. | 
THE equation of the surface is 
W = (aa? + By? + 2") (a? + +2) -C#(P+e)eP-B(e+e0)¥ 
-¢(¢ +b’) 3?+ ab’ =0, 
and, by common algebra, this equation may be written in the form 
Jee ar Libb&= 0, 
‘where P stands for 
au? + Dy? + c's? +b? (a + y* + 87) — Bc + a’), 
and Z, for a / 0-0 +8/P—-O+bf/ae-e, 
7, for 0 / @-0-3/P-C+b/a-e, 
I, for tf &-0 +38 /B-8-b /a—c, 
and i, for—2@ /@-B+2fP-8+b fae-e; 
hence, it follows at sight, that the plane /, = 0 meets the surface in 
a conic, and touches the surface all along this conic, or else the 
1 The perpendicular from the origin on the plane /; = 0 is 6, and the perpendicular 
on the parallel tangent plane to P= 0 is 
d 204 (2 + a?) 
(a? + B)(b? + 0?) ’ 
the difference of the squares of these perpendiculars, the latter being the greater, is 
b2 (a2 — B) (62 — c) 
(a? + 6) (6? + ¢*) ? 
which shows the conic is real, as a>b>e. 
