PROFESSOE C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 317 
because they are independent of r; the lines of the former cono-mencv are double 
edges and double tangents. 
We proceed to determine the class of the fociil surface. The equations 
{pqah) = 0, = Q .(546) 
require a ray to intersect the fixed line a, h. Eliminating p, the equation of the 
locus of q is 
/(TT) + T/= 0, or [/(qf/),/(ap),/(6q)] . . (547); 
and this (274) is a curve of order = H and rank v — 48. But this curve is a 
complex curve consisting of the line ah and a residual which intersects it in four 
points on the Jacobian. The order and rank of the residual are m— 10, r = 40, 
the rank being diminished by twice the number of intersections. The number (r) of 
tangent planes through ah to this curve miuus twice the number of intersections 
gives the number of planes tlirough ah containing consecutive rays. Thus the class 
of the focal surface is N = 32, and its order (524) is M = 48. Every one of these 
numbers is double the corresponding numlier obtained in Art. 142 for the perniutable 
function. 
For the sake of completeness we wisli to show the nature of the assemblage of 
lines common to the complex (538) and the second complex (541), as we have already 
completely considered the lines common to the remaining two pairs. Evidently the 
congruency of hi-connectors belongs to these two complexes and is counted twice 
among their common lines. 1 here remains an assemblage of lines of order 
p = ‘i X G 2x 5 = 8, and of class p = '3 X G — 2 X h = 0. It is easy to prove by 
the method of this article tliat these lines join an arbitrary point to the eiglit 
correspondents of r in the quadratic transformation /( j>i>) = r. 
SECTION XX. 
The Method of Arrays, 
Applications to n-Sijstems of Linear Functions. 
Art. 
150. The expansion of arrays and the determination of the scalar coefficients. 
151. Conditions that a function of an «-system should convert m quaternions into m others 
152. Conditions for the conversion of m points into m others. 
153. Conversion of lines and planes into others. 
154. Relations connecting points with their transformeds when conditions must he satisfied. 
The three types. 
155. The critical systems for functions of an w-system. The four types. 
156. Conditions that a line may he destroyed hy a single function of an «-systcm . . . 
15/. Conditions that a line may he destroyed point hy point hy an included ra-system . 
158. The various methods of destroying a plane. The destruction of a hyperboloid 
generator hy generator . 
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