8 ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 
tion (12) therefore expresses a projective transformation T., 
which transforms the point « directly to x,, and is equivalent 
to the successive applications of T and T, in the order named. 
The transformation T, is called the resultant of the trans- 
formations T and T,, which are called the component trans- 
formations. The operation is symbolized thus: TT,=T.. If 
the two component transformations 7 and T, are taken in the 
reverse order, the resultant, T7,7=T,’, is not the same as T.. 
Thus: 
aera (aa+be1) «+ (abi+bd;) 
r, aloe (cai+de,) «+ (ebi+dd)) ’ (12’) 
which is not the same as T,. The two projective transforma- 
tions T, and T,’ are called conjugate transformations. 
By referring to the transformations lettered T, T,, T., we 
see that the determinant of 7, is 
aat+bie ab+bhid| , 
aat+die cb+did}? 
but this is the product of 
a b |ay bi | , 
(c 4 by }¢1 di}? 
these determinants are respectively the determinants of T and 
T,, the components of T,. Hence the determinant of a trans- 
formation, 7T,, which is the resultant of transformations T 
and T,, is equal to the product of the determinants of T and T,. 
This result is capable of immediate extension; for let T,, 
T, and T, denote three transformations, the result of whose 
successive applications is equivalent to 7; the compounding 
of T, and 7, is equivalent to a third transformation, T,,. 
The resultant of 7, and T. is T,, and the determinant of T, is 
equal to the product of the determinants of T,,, and T.; hence 
the determinant of T, is equal to the product of those of T,, 
T,, and T,. This mode of reasoning is applicable to the re- 
sultant of any number of transformations; hence by induc- 
tion we infer the following theorem : 
THEOREM 6. The resultant T, of % projective transformations 
fh. (Cit se ly eae oe cer n-1) is a projective transformation, and the 
determinant of the resultant is equal to the product of the determin- 
ants of the components. 
