216 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
Hence, assuming that the 2 m lines each counts 6 times*, 
Order of surface =2 m 
Order of tiecnode surface =ll(2m— 24) or 22m —24 
Order of intersection 
Nodal curve, i ^(m 2 — -m)+$ circles, 11 times 
Cuspidal curve * circles, 27 times 
Circles of contact m 2 — m — 2^ — 3*, 
Lines of contact 2m , 6 times 
Elecnodal curve, 5m 2 — 9m— 10<5— 12* circles each twice 
= 44m 2 — 48m 
22m 2 — 22m+44S 
+ 54 * 
2m 2 — 2m— 4S— 6* 
+12m 
20m 2 — 36m — 40S — 48* 
44m 2 — 48m 
The Flecnodal Torse. Article Nos. 48 & 49. 
48. Starting from 
22V + 2 Id = 6 (65' + 8d) — 7(25' + 3c') 
= 6(3 n 12 — 6 n' — *) — 7 (n’ 2 — n' — ei) 
=lW 2 — 29»' + 7-6*, 
that is 
llw' 2 — 24w'— 225'— 27c'=5w'— 7S+6*, 
I find 
%'(lL*'-24)-226'-27c' 
=?t(w— l)(lLt— 24) +6(—59?a+ 96)+c(— 94^+156) + 26& 2 +87e 2 
— 52# — 114#+ 141/3 + 94y + 77«’+ 3 j + 4% 
— 150— 45^— 10C— 9B. 
49. For a surface of the order n without singularities this equation is 
ri(lln'-24:)-22V-27c'=n(n-l)(ll7i-2i); 
to explain the meaning of it, I say that the reciprocal of a flecnode is a flecnodal plane, 
and vice versa : the reciprocal of the flecnodal torse of the surface n (viz. the torse gene- 
rated by the flecnodal planes of the surface) is thus the flecnodal curve of the reciprocal 
surface n' ; and the class of the torse must therefore be equal to the order of the curve. 
The flecnodal torse is generated by the tangents of the surface n along the curve 
of intersection with a surface of the order Yin — 24 ; the number of tangent planes 
which pass through an arbitrary point, or class of the torse, is at once found to be 
n{n—l)(Yln — 24) ; for the reciprocal surface the order of the flecnodal curve is by what 
precedes n'(lln'~ 24)— 225'— 27c'; and the equation thus expresses that the order of 
the curve is equal to the class of the torse. 
* See last foot note : the same remark applies to the present terms in m, which cannot be got rid. of by an 
alteration of the numbers 22 and 27 to which the investigation relates. 
