352 A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. [411 
all satisfied if only 
h' = 5, 
g’ + 16i/ = - 138, 
g' + = 38, 
9 +9' = 18, 
X =-1. 
62. I remark however that the cubic scroll XXII or XXIII gives 
0 = 54 - (330 - 272) - 2 (X + X'), 
that is, X + X' = —2, instead of X + X' = —8. The investigation is in fact really in 
applicable to a scroll, for every point of a scroll has the property of a flecnode; 
whence if U = 0 be the equation of the scroll, that of the flecnodal surface is 
M. U = 0, containing U as a factor, and there is not any definite curve of inter 
section constituting the flecnodal curve; but I am nevertheless surprised at the 
numerical contradiction. 
63. Combining the two sets of results, we find 
h =24, 
9 =9> 
x — x, 
X =- 1, 
/a = 10 + \g, 
v —¥ + t\9> 
K = 5, 
/ = 18 —<?, 
x' = x, 
X' = - 7, 
/ = 6- \g, 
~-h9\ 
and the formula thus is 
/3'= 2n(n — 2)(ll?i — 24) 
— (110w — 272) b + 44<£ 
- (116w- 303) c + ^r 
+ + 248 7 + 198i 
- 24(7+j - 10 x + *2-0 - 5(7' -185' - 6*' - £0' 
— xi — x'i! + -fag (— 165 — 8% — d + 165' — 8 X — O'),