218 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
h' =\n{n- 2)(16w 4 - 64w 3 + 80^ 2 - 108»+156.), 
r' =2n(n— 2){3n—4), 
d —4n{n— 2), 
5' =0, 
x'=o, 
0 = 0 , 
B'=0, 
ft' =2n(n-2)(lln-24), 
y' =4n(n-2)(n—3)(n 3 —3n-j-16), 
i! = 0. 
Investigation of Formula for ft'. Article Nos. 51 to 64. 
51. The value ft'=2n(n— 2)(lln— 24) for a surface without singularities was obtained 
by Salmon by independent geometrical considerations, viz. he obtains 
2ft'=4n(n-2)(lln-24) 
as the number of intersections of the spinode curve (order =4n(n— 2)) by the flecnode 
surface of the order 11 n — 24. 
52. The value of ft 1 must be obtainable in the case of a surface with singularities, 
and I have been led to conclude that we have 
ft'= 2n(n-2)(lln-24) 
— (110^—272)5+ 44^ 
-(116?i-303)c+^r 
+£f+3 + 248 7 +1985 
+ linear function (i, j, 0, %, C, B, i',j', C', 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 n without singularities, i,j, 6, %, C, B, 
O, B' then all vanishing, is the identity 
0= 2n'(n'-2)(lln'-24) 
— (1 1 — 2 72)5' + 44c[ 
-(116^-303)c'+^ 
+ “f i /^ , + 248y'+198^ * 
(n', 5', (f c', r', ft', y, t' having the values in the foregoing Table). It was by assuming 
for ft an expression of the above form but with indeterminate coefficients, and then de- 
termining these in such wise that the reciprocal equation should be an identity, that 
the foregoing formula for ft' was arrived at. 
