450] BETWEEN TWO PLANES, AND ON SPECIAL SYSTEMS OF POINTS. 255 
lines; viz., “a 1 = 12, — % We may have 12 points and 2 points such that, for a 
quintic curve passing once through each of the 12 points and twice through each of 
the 2 points, the number of conditions actually imposed (instead of being 12 + 3.2, =18) 
is = 17.” The construction is as follows: viz., starting with the 2 points and any 
10 points, we may draw a quartic passing twice through the first of the 2 points, 
once through the second of them, and through the 10 points; and another quartic 
passing twice through the second of the 2 points, once through the first of them, and 
through the 10 points: the two quartics intersect in the 2 points each twice, in the 
10 points, and in 2 new points, forming, with the 10 points, a system of 12 points; 
and the first-mentioned 2 points and the 12 points form the system in question. 
A more complicated case, ^ = 10, a. 2 — 6, a 3 = 1, occurs in Dr Nother’s paper, “Ueber 
Flachen, welche Schaaren rationaler Curven besitzen,” [Math. Ann., t. nr. (1871), pp. 
101—227]. Except these two, I do not know any other case of a special system for 
which a 2 , a.,... are not all =0; the investigation of such systems would, I think, be very 
interesting. 
[A concluding paragraph of seven lines gave some corrections to the “ Memoir on 
the Rational Transformation between Two Spaces,” 447, which corrections are made in 
the present reprint of that paper.]