322 PROFESSOE C. J. JOEY OX QUATERNIOXS AXD PROJECTIVE GEOMETRY. 
functions of a ten-system. For a fourteen-system it requires an invariant relation 
to vanish. 
IV. This case requires a single function to destroy a point ; it gives the lines 
destroyed by functions of a five-system (of these there are 20, compare Art. 114); 
and it imposes a condition on a nine-system of functions, so that some function of the 
system may be capable of destroying a plane. For a thirteen-system an invariant 
relation must vanish if a critical case arises for non-coplanar points. 
I calculate the order of the Kummer surface of the quartic complex for the eight- 
system to be 72, and the order and class of the congruency of the double lines to be 
24. The lines of this congruency would seem to be capable of being destroyed by 
two-systems of functions selected from the eight-system. 
156. More particularly, if the line ah can be destroyed by a single function of an 
n-system, 
tx^f\a = 0, = 0 .(565); 
and the array 
/i« /s« ■ 
• • • /«& -J 
(566) 
must vanish. The number of conditions is now 9 — n, so that from a nine-system 
one function can be found to destroy an arbitrary line. For n = 8, we have the 
complex 
S ± ifiaf.af^af^a) = 0 .(567). 
If the plane a, h, c can be destroyed by a single function 
' J\a f„a...f,,a' 
fi^ - ■ ■ fnh !> = 0 . . 
/a /sC . . . f,fi 
(568), 
and this requires 13 — n conditions. For n = 12 we have the surface enveloped by 
the plane (compare the last article) 
S ± (/i«yw>/#) (/5^/0^/7^/s^) (/gc/ioc/nC’/iaC) = 0 
(569). 
157. When a line can be destroyed point by point by functions of a two-system 
selected from an ?i-system, 
S{x^-\-ty^)f^{a-\-th) = 0, or Sxj\a = 0, Sxj\h+Syj\a = 0, tijj\b = 0 (570); 
and the array 
f^a . . . f„a 0 0 ... 0 
■ • • f>P ./i« ./> . • • |> = 0 . . 
(571) 
0 0 ... 0 f,h . . . J 
must vanish, or 13 — 2)i conditions must he satisfied when the line is arbitrary. The 
