318 PROFESSOR C. J. JOEY OX QUATERNIONS AND PROJECTIVE GEOMETRY. 
150. We shall illustrate the method of quaterniou arrays* hy a few examples on 
systems of linear functions. These functions may be supposed to be of the most 
general kind, functions of a point in sjaace of p, dimensions, but we pay particular 
attention to the case of three dimensions. 
An array of n rows and m columns vanishes if, and only if, the constituents in the 
rows are connected by tlie same set of scalar coefficients Xo . . . x^. Thus 
a 
2 Ci-o 
h.y h,. 
/j 4 4 
Vi Ih V-i 
r, 7-3 
when 
a„ 
jV 
= 0 
(548), 
tx,a, = 0 , tx,bs =0, . . . tx,r, = 0 . 
It is proved in the memoir that the expansion of the array is of the forint 
(549). 
^ ± {bJ)J),hs) . . . (/ 4 „-_ 3 , hn'-2, hn'-^, kn) 
^ Vin'+\ + - ■ ■ V“' 1 
' . . . i 
I, 
^4n'4-2 • • ^ , 
(550) ; 
and we take definitely m = Vn'-\-n”, where = 0, 1, 2 or 3. The number of 
equivalent scalar conditions is 4m — n 1 for the vanishing of a quateinion array, 
and (/r + 1) m — 77 + 1 for an array of jioints in p. dimensions. 
The scalars x^, &c., are determined 'when (548) is satisfied by the system of 
arrays of m — 1 columns and n inws, of which tliis array 
is a type. 
f X^a^ + X.Xi:„, «g, 
I + xj)^, + 
I 
.apq + x,ro, 7-3 
• • «;« 1 
h. ! 
. (551) 
^ ‘Trans. Roy. Irish Acad.,’ \nl. 32, jip. 17-30. 
T Every row must lie represented in the expansion, and it may he gathered from the Memoir how to , 
expand if one row involves only four constituents. In this case the general method fails. 
