18 TYPES AND NORMAL FORMS. 
denoted by TS or ST according to the order in which they 
operate. The resultant of T and Tis denoted by 7. If T 
be repeated n times the resultant is denoted by T”, etc. The 
resultant of T”™ and T” is T”*". The resultant of T and its 
inverse is denoted by T7T-1=7’°=1. Thus unity is a con- 
venient symbol for the identical transformation. 
A symbolic equation of the form, 
HS — Sh e@s 
means that the resultant of the three transformations on the 
left is equal to that of the five on the right when taken in 
the order indicated. We may multiply each side of this 
equation by say 7’ by writing T~’ before each side; thus 
eT Si — le Melee 
Suppose we have two symbolic equations such as 
SS LS IE Ba Sy SIL 
by multiplication we get 
SESH) = SIPS i ie Syl ILS Side 
These examples sufficiently illustrate the principle. 
24. Operation on S by T. If TS is not the same transfor- 
mation as ST let us assume that there exists a transformation 
S’ such that TS’= ST. Let us multiply both sides of this 
equation by 7’, the inverse of T. This gives us T ‘TS’ = 
S’= TST, since TT~ is the identical transformation and 
denoted by unity. Hence S’ is a projective transformation, 
since it is the resultant of T7-", S and 7, each of which isa 
projective transformation. 
The two transformations S and S’ are conjugate transfor- 
mations, as the following equations show: S’=(T4S)T and 
S=T7T(T"S), i. e., (T-1S) and T combined in one order give 
S’, and in the reverse order give S. 
We say that the transformation S’ is obtained by operating 
on S with 7. Thus the operation of T on S produces S’ and 
is symbolized by S’'= T-'ST. This equation may be solved 
for S, so to speak, by the following process: Write T before 
each side of S’= T-'*ST and we get TS’=ST. Now write 
