PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOxMETRY. 319 
It all the minor arrays formed by omitting one column of (548) vanish, we take any 
two of these minors, and forming second minors coi'responding to (551) we obtain two 
sets of relations (549), and so on in general. 
151. In order to find the conditions that a linear function of an ii-system should 
convert m given 'iceighted points rtj, . . . into m others, . . . b„„ we write down 
the array in m rov^s and a + 1 columns, 
1 /d'-\ bM\ ■ 
• Uh 
Id 1 
1 / I'C ■ 
\ 
■ /AC 
! 
1 
1 
r 
1 
• J 
1 / 
1 • 
1 
C. J 
= 0 . 
(552), 
whose vanishing requires 
.( 553 ). 
Ihe vanishing of this array requires 4?u — n scalar equations to he satisfied. If 
then n =■ 4ai, the array vanishes without restriction, and a single condition must Ije 
satisfied for the vanishing of the arrays, such as (551), 
I 1 
! ‘^‘1/1% ■!" • ■ .Ajq ! 
= 0, &c. . 
. (554), 
I d- h,,, j 
and these determine the coefficients x without ambiguity. 
Thus from a given 4j?i-system can he found one function wliicli shall convert in 
given weighted points into other given weighted points. (Compare Art. 3.) 
152. When the weights are disregarded, the equations of.condition are 
%xj,a^=-. Sxjla. = g^b.2, . . . txJ/A„ = yJ,,, . . . ( 555 ); 
and these furnish the array 
{ fi"i /Yh • • • />! />! 0 U . . 0 1 
! /otto . . . f,a-, 0 b., 0 . . 0 j 
..0. . . . (556), 
I 0 . U 0 . .b,„} 
of -f- n columns and m rows. Its vanishing requires 3//i - h + 1 conditions to be 
satisfied, and the vanishing of the minor arrays such as (551) requires a single 
condition if n = 3?a fi- 1, and these definitely determine the function. Thus from 
a (Sm l)-system can be found one function which converts 7 ii points to m others 
when the weights are neglected. In particular, a linear transformation can he found 
(out of the whole sixteen-system) to convert five points into five others (Art. 3). 
