514 
Proceedings of Boyal Society of Edinburgh. 
“Equations donnees. 
ax +hy +CZ + dt =f 
a'x + b'y + c'z + d't =/' 
a"x +//V +c"z -\-d"t = f" 
a^"x + h"'y + c'”z + d"'t=f". 
Calcul. 
ah' 
1 
II 
P 
ah" 
11 
1 
ah" 
1 
II 
ah'" 
^hcd" = 8, 
a!h'" 
-h'a'"= €; 
m = c"a 
- c'/3 + cy , 
n — d"cL — c'S + ce , 
m'=f"a -f'P+fy, 
«' =f"'a-fS +/c; 
- Sd' + €d) - n{ad" - pd' + yS) 
}■ 
N'" = ^(^mi - nm! ^ , 
N" = 1 1 - M + cfO - Til {ad!’ - 13d' + yd) I , 
fD-cW-d^'" =S, 
fB-c'K'-d'W" = S' , 
_ aS' - Sa' 
a 
S&' - 5S' 
a 
) 
N' 
D 
D 
N'" 
‘-T-" 
The explanation of the mode of procedure is not difficult to see. 
(1) The determinants \ah '\ , \ah "\ , \ah"\ , \ab'"\ , \a'h'" are calculated. 
(2) With the help of these are next got four of a higher order, 
viz. \ah'c "\ , \ab'c "\ , \ah'f "\ , \aVf '"\ . 
(3) Two others of the same order, viz. 
ad'" - M + €fZ, ad" - /3d' + yd 
% 6 
\ah'd"'\, \ah'd"\, 
