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