PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
223 
thing, apply the original equation to the reciprocal surfaces, as given by interchanging 
the upper and lower halves of the Table of Singularities, we have another series of equa- 
tions, viz. this is 
0=54432-54432, 
(I) 
0=27851-27846- hi. 
7/ =5, 
(H) 
0=18180-18318- g'- 16*, 
y+16*= — 138, 
(III) 
0=11765— 11756— 2//— a, 
27/ -(- A = 9, 
(IV) 
0= 6917- 6584- h'-g'-p- 8*/ 
hi +g* + y>~\- 8* =45, 
(VI) 
0 = 35 34 - 35 22 — 37/— 3a, 
37/+ 3A =12, 
(VIII) 
0= 3024- 3144— 2/— 16*— 
2#'+16*+y=-120, 
(IX) 
0= 1433- 1386-2h'-g'-X-2[x, 
2A'+^ , +A+2 i M/=47, 
(XIII) 
0= 518- 504— 47/— 6A, 
47/ + 6a =14, 
(XVI) 
0= 383- 322- h'-2g'-g-2(*, 
7/ +%'+</ + 2g,=61, 
(XVII) 
0= 54 —3g—3g', 
3^+3/ =54, 
(XXI) 
satisfied if only 
7/ = 
5, 
</ + 16*= 
-138, 
g' + 2y, = 
38, 
9+9' = 
18, 
A = 
- 1. 
62. I remark however that the cubic scroll XXII or XXIII gives 
0=54-(330-272)-2(a+a'), 
that is, A+A'= — 2, instead of A+A'= — 8. The investigation is in fact really inappli- 
cable 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 intersection 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 =ff> 
x —x, 
A = — 1, 
t*= 10 +&, 
' =-¥+T^, 
7/ = 5, 
