214 THEORY OF COLLINEATIONS. 
is rational, except when the path-curves are straight lines or 
conics. 
For any integral value of 7, except the special values 
1, 0, ©, —1, 2, there are six subgroups for which the 
path-curves are algebraic curves of the order r, viz.: 7, 1/r, 
1—r, 1/(1—7), r/(r—1), (r—1)/r. Thus for r=32 the 
path-curves are cubics, viz.: 
je (OK? 
the path-curves are also cubics for r= 1/3, —2, —1/2, 2/3, 
3/2, and for no other values of r. 
256. Involutoric Collineations in G,(AA’A”). The 
group G,(AA’A”) contains three subgroups of perspective 
collineations, viz.: whenr=1,0, ~. Each of these groups 
of perspective collineations contains an involutoric collineation 
and hence G,(AA’A”’) contains three distinct involutoric col- 
lineations. The values of k and k’ which correspond to these 
involutorice collineations are as follows: When k= —1 and 
k’ = — 1, the involutoric collineation has its vertex at A and 
A’ A” for axis; for the pair of values (— 1, 1) the collineation 
is involutoric, having A’ for vertex and A A” for axis; for 
the pair of values (1, — 1) the collineation is again involutoric 
having A” for vertex and A A’ for axis. 
The equation kt=~*k” is satisfied by the pair of values 
k = —1, k”=1 when r is rational with even numerator and 
odd denominator; it is satisfied by the pair of values (—1, —1) 
when r is rational with odd numerator and odd denominator; 
it is satisfied by the pair of values (1, — 7) when r is rational 
with odd numerator and even denominator. The equation is 
not satisfied by either of these pairs of values when ¢ is an 
irrational number. Hence the involutoric collineation given 
by the pair of values (— 1, 1) belongs to every subgroup of 
G,(AA’'A") for which r is rational with even numerator and 
odd denominator; the involutoric collineation given by the 
pairs of values (— 1, —1) and (1, —1) are contained respec- 
tively in every subgroup for which r is rational with odd nu- 
