PROFESSOE e. J, JOEY OX QUATEliNIONS AND PROJECTIVE GEOMETRY. 323 
fundions must satisfy 9 - 2» conditions, as the line may be made to satisfy four. 
For a four-system one condition must be satisfied for the existence of a line of this 
nature, but for a five-system (compare Art. 114) a ruled surface of such lines exists, 
triple chords of a curve of the tenth order. 
If the line can be destroyed by functions of a three-system we have (compare 
Art. 114) ^ 
S (.T^ -f {a tl)) = Q .(572), 
and the resulting array is of 4 rows and 3n columns, and vanishes if 13 — 3n 
wnditions are satisfied. Finally, if the line is destroyed seriatim by functions of an 
included four-system, 21 — 4'n conditions must be satisfied. 
We may state that the number of conditions required to determine an N-system 
included in an ^r-system is 
N [n N) _ N' (n — (N fi- N' = w).(573). 
158. As regards the destruction of planes, a plane may be destroyed en Hoc, as 
in (568), 01 line by line, or point by point. In the second case, 
% (xj -}- syT^f-^ (rt -}- 
or t (Xj + sy^)j-^ [a + sc) = 0, S (x. -f sy^) {b sd) = 0 . . (574), 
with the condition [abed) = 0 . 
Thus the array is 
/i« 
/f 
0 
/l& 
/ 2 « 
/ 2 C 
0 
Ad 
0 0 
fnC 
0 
fnd 
0 
0 
/l« 
/f 
0 
/i&. 
Ad- 
. . 0 
■ • /«« 
• Ac 
. 0 
-A^ 
. Ad 
^ = 0 
(575) 
of 6 rovs and 2n columns, requiring 25 —2n conditions when we disregard [abcd) = (). 
This IS the case m which a function can destroy a hyperboloid* generator by 
generator. The same number of conditions must be satisfied even when the four 
points are supposed co-planar. 
Finally, the case m winch the points are destroyed seriatim gives an array of 3l^ 
columns and 6 rows, requiring 25 - 3n conditions for its vanishing. 
From these articles we can clearly trace the way in which a Jacobian of four 
uncDons may degiade, one of the most interesting being where it breaks up into 
a pair of quadrics, one of which is destroyed generator by generator by a two- 
system. 
_ * In the paper on the interpretation of a quaternion as a point symbol, the equation q = a + th + sc + std 
IS considered. It represents a ruled quadric and exhibits the dual generation. 
2 T 2 
