CONTINUOUS GROUPS. 135 
where the three functions f, ¢ and ¥ are linear and homoge- 
neous inw, yand z. TJ transforms the point (a, y,z) into the 
point (“,, y;,2,). Let T, be a second collineation of the same 
system which transforms the point (w,, y,,2,) into the point 
(2, Yz,2.). The equations of T, are 
TE 5 pee Ba ay a a (3’) 
P)22 = Wn (X41, Y1,21); 
where the functions f,, ~, and v, are of precisely the same 
form as for T'; they differ only in the values of the coefficients. 
The resultant, T,, is found by eliminating ~,, y,, z, from the 
two sets of equations. This gives us a set of equations ex- 
pressing %,, Y,, 2, directly in terms of a, y, z. T, may be 
written 
P2%2 = fe( x,y,z), 
ie : eat ee ie (3”’) 
Po z =wWe (x,y,z). 
If the functions f,, ¢., and ¥, are of the same form as 
the corresponding functions for T and T, and differ only in 
the values of the coefficients, then 7, belongs to the same 
system as T and T, and the system possesses the first group 
property. 
168. An Illustrative Example. As a simple illustration let 
us consider the system of collineations of type II given by 
equations (26), Art. 149. Let T be given by the equations 
Pmu=x+tz, 
DT by ky, (4) 
Paz=2Z. 
Let T, be given by the equations 
awv=u+hz, 
ths > Ay=hy, (4’) 
Pl 22 = 21. 
Eliminating (2,, y;, z,) from T and T, we get T, as follows: 
poawe=ax+(t+h)z, 
2: pye=khy, (5) 
f2%2=2, 
which is of the same form as J and T,,. 
