PEOFESSOE C. J, JOEY ON QUATEENIONS AND PEOJECTIVE GEOMETEY. 321 
In tyjDa T, if is given, h„i lies in a plane; and cim lies on a general cubic surface 
if h„, is given. 
In type II, if a„, is given, h,„ may be any point on a line ; and if is given, 
may be any point on a twisted cubic. 
In the third case, and either point is determined it the other is given. 
There is no difficulty in applying this method to the case of Art. 153. We must, 
however, include the case of four conditions being recpiisite. The last line must 
belong to a complex, a congruency, a ruled surface, or be one of a definite number 
of lines. 
155. We shall now consider the critical cases when eveiy first minor of (552) 
vanishes. 
The minor obtained by omitting the last column expands into 
- ± (/i(t<^i),/o(«i),/3(^h)j/r(«])) • • • [/n«-3 (n,„). . . (5G2). 
Here, as in the last article, we have the types 
I- (Fpf-,;;, FoCO;;, FgCt,,;, 0 
II. FgU,,,, FgU J = 0 ; 
III. [F^a,,, FoCgJ =0 ; 
IV. F«,;, = 0. 
corresponding to n — 4to, n - 4m + 1, n = 4m + 2, and n — 4m + 3. 
Now, fiom the nature of arrays, though it does not appear directly from the form 
of the expansion, these conditions are all combinatorial functions of the m points a. 
I. In the first place, for the type I we have for m = 1 the Jacobian of a four- 
system. Next, for n = m — 2 we have a one-conditioned assemlilage of lines of 
the fourth order, or a complex of the fourth order. These are the lines which can be 
destroyed by single functions of the system. For n = 12, m = 3, (562) represents a 
one-conditioned assemblage of planes, and these planes envelope a surface of the 
fourth class, and eacli can be destroyed by a corresponding definite function of the 
system. 
For n= 16, m = 4, the same ecjiiation represents a constant multiplied by the 
volume of tlie tetrahedron (■''h'^GUs^) to the fourth j^ower. 
II. Again, for n = 4m — 1, and more particularly for m = 1, we have the critical 
sextic 
L/i«/2A/3^^] = 0.(564), 
of three functions ; and for seven functions a congruency of lines common to a set of 
quartic complexes ; while for eleven functions we have a two-conditioned assemblage 
of planes, or a developable of planes enveloping certain surfaces of the fourth class. 
III. For 71 = 4 HI 2 there is first the system of united points of for a pair 
of functions, or = 0. Secondly, a ruled surface of lines destroyed by 
functions of a six-system ; and thirdly, a determinate number of jDlanes destroyed by 
2 T 
VOL. CCI.—A. 
