411] 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
345 
Article Nos. 48 and 49. The Flecnodcil Torse. 
48. Starting from 
22h' + 27c' = 6 (66' + 8c') - 7 (26' + 3c') 
= 6 (3ft' 2 - 6»' -k) — 7 (ft' 2 - n f - S) 
= lln' 2 — 29??' + 7 — 6«, 
that is 
lift' 2 - 24ft' - 226' - 27c' = 5ft' - 7S + 6«, 
I find 
ft'(lift' - 24)-226'-27c' 
= ft (ft -1) (lift - 24) + 6 (- 59ft + 96) + c (- 94ft + 156) + 266 2 + 87c 2 
- 52k - 114k + 141/3 + 94 7 + 77% + 3j + 4 % - 156» - 456 - 10(7 - 9B. 
49. For a surface of the order n without singularities this equation is 
ft' (lift' - 24) - 226' - 27c' = ft (ft - 1) (lift - 24) ; 
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 
generated by the flecnodal planes of the surface) is thus the flecnodal curve of the 
reciprocal surface ft'; 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 lift —24; the number of tangent 
planes which pass through an arbitrary point, or class of the torse, is at once found 
to be ft (ft — 1) (ll?i — 24); for the reciprocal surface the order of the flecnodal curve 
is by what precedes ft'(lift'— 24) — 226'— 27c'; and the equation thus expresses that 
the order of the curve is equal to the class of the torse. 
Article No. 50. The general Surface of the Order n without Singularities. 
50. In the general surface of the order n without singularities, we have 
n = ft, 
a = ft 2 — ft, 
8 = \n (ft — 1) (ft — 2) (n — 3), 
k = ft (ft — 1) (n — 2), 
6=0, 
k =0, 
6 =0, 
2=0, 
C. VI. 
44