TYPES AND NORMAL FORMS. 307 
This resultant will be of type II when the condition, (a +d)’ 
— 4(ad — bc) = 0, (art. 12) is satisfied. The determinant of 
T,/ is equal to the product of the determinants of 7’ and T,/ 
which are each equal to 7, hence A, = ¢ dl = 1, and the above 
condition reduces to (a+d)?=4. Applying this condition 
to equation (27) we get in terms of their natural parameters 
the condition that the resultant of TJ’ and T,/ shall also be of 
type II, viz. : 
it,( A — A,) : tt,(A i An) = 4 f = 0. 
Hence we must have one of the four following cases: t = 0, 
fi (oA — Aor ti (AIHA, = 4: 
These necessary conditions are also sufficient. If t=0 or 
t, = 0, then T” or T,’ is the identical transformation. The re- 
sultant of any transformation T and the identical transforma- 
tion is evidently 7. If A,= A, the two transformations of 
type II have the same invariant point. Sufficiency of this con- 
dition is shown in (art. 28). Finally let any numerical values 
be assigned to A,, A, ¢,, and t such that tt,(A —A,)*=4. 
For example let A= 4, t=i1and A,=2,t,=1. T”’ reduces 
5a — 16 ey, : 
(emi, becomes! ¢,—— =~.) 7, "istfound to be 
eine Navas? 2 whicn 1S 0 ype ’ Soe an a Ae 
Considering the identical transformation as not properly of 
type II, we reach the following result : 
THEOREM 10. The necessary and sufficient conditions, that the 
resultant of two transformations of type IL should also be of type 
II, are (1), that they have the same invariant point; or (2), that 
tt,(A —A,)*= 4. 
23. Symbolic Notation and Operation. A very useful and 
convenient symbolism has been invented for dealing with 
certain transformations and their combinations. We proceed 
to explain and illustrate this notation. 
A transformation is denoted by a single letter Tor S. The 
inverse of Tis denoted by JT. The resultant of T and S is 
=P 
