411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 347
Article Nos. 51 to 64. Investigation of Formula for ¡3'.
51. The value /3' = 2% (% — 2) (11% — 24) for a surface without singularities was
obtained by Salmon by independent geometrical considerations, viz. he obtains
2/3' = 4%(%-2) (11% — 24)
as the number of intersections of the spinode curve (order = 4n (n — 2)) by the flecnode
surface of the order ll?i — 24.
52. The value of /3' must be obtainable in the case of a surface with singularities,
and I have been led to conclude that we have
/3'= 2%(% — 2)(ll?i — 24)
— (110% — 272)7> + 44g
— (116%-303) c + ^r
+ 69i/3 + 248y+198£
+ linear function (i, j, 0, y, G, B, i!, f, O', f, B),
but I have not yet completely determined the coefficients of the linear function. The
reciprocal formula in the case of a surface of the order % without singularities,
i> 3> X> B, 7, j', O', G', B' then all vanishing, is the identity
0= 2% / (%'-2)(ll% , -24)
— (110?i' — 272) b' + 44^'
-(116%'-303) c' + ^r'
+ ^/3' + 248 7 ' + 198i'
( n b', q, c', r', /3', f, t’ having the values in the foregoing Table). It was by assuming
for /3 an expression of the above form but with indeterminate coefficients, and then
determining these in such wise that the reciprocal equation should be an identity,
that the foregoing formula for /3' was arrived at.
53. I assume
/3'= 2%(%-2)(ll%-24)
-b(An-B) + Cq
— c (D% — E) + Fr
— G/3 — //7 — It
+ linear function (i, j, 0, G, B, %, f, O', G', B'),
where it is to be remarked that, in virtue of the equations obtained No. 11, two of
the coefficients of this form are really arbitrary: I cannot recall the considerations
which led me to write D = 116, E = 303.
44—2