523J 
ON THE TRANSFORMATION OF UNICURSAL SURFACES. 
393 
formation of the plane figure, then we have co = 0. I presume that for the most 
simple transformation, that is, when n has its least value, %a r has also its least value, 
and consequently that w is =0. 
Recapitulating, the results obtained are 
q = — 2(n— 13) b + n s — 14n 2 + 31 n —18 + 4 (%a r — to), 
t= (n— 8 )b — ^n 3 + 4>n 2 — %£-n + 5— (Sa r — to), 
where it will be recollected that 
b = ^ (n — 2) (n — 3) + © ; 
the formulae are verified in the several cases: 
n' 
< 
n 
e 
CO 
b 
Q 
t 
2 
2 
0 
2 
0 
1 
0 
0 
0 
Quadric surface 
2 
1 
0 
3 
1 
0 
1 
0 
0 
Cubic scroll 
2 
0 
0 
4 
2 
0 
3 
0 
1 
Steiner’s quartic surface 
3 
6 
0 
3 
0 
0 
0 
0 
0 
Cubic surface 
3 
5 
0 
4 
1 
0 
2 
2 
0 
Quartic with nodal conic 
2 
8 
1 
4 
0 
0 
1 
0 
0 
Do. with nodal line 
2 
7 
1 
5 
1 
0 
4 
8 
0 
Quintic with nodal quadriquadric 
2 
11 
0 
5 
0 
0 
3 
4 
0 
Do. with nodal skew cubic 
2 
12 
2 
5 
-1 
0 
2 
0 
0 
Do. with two non-intersecting nodal lines 
which are the transformations chiefly as yet examined: but the first-mentioned case 
(quadric surface, generalised stereographic projection), although as stated the formulae 
are verified with the value to = 1, does not really come under the foregoing theory. It is 
interesting to see that they are verified in the last-mentioned case, belonging to a 
negative value of ©, that is, to a special system of fixed points. 
Cambridge, Dec. 5, 1870.