NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 



117 



called the resultant of the transformations T and Ti, which are called 

 the component transformations. The operation is symbolized thus: 

 TTi = T2. If the two component transformations T and Ti are taken in 

 the inverse order, the resultant, TiT=T'2, is not the same as T2. Thus : 



rrv . „ , _(aa, + bc,)x + (ab t -f bd,) 



X2 



(3) 



(caj + dc,) x + (cb, + dd,) 



which is not the same as T>. The two projective transformations T2 

 and T'2 are called conjugate transformations. 



Determinant of a transformation. — The determinant 



b 



A = 



is called the determinant or modulus of the transformation (1 ), and 

 it is assumed, for the present at least, that A does not vanish. 



By referring to the transformations lettered T, Ti, T2, we see that 

 the determinant of T2 is 



a x a + b,c ajb + b t d 



c x a + d,c Cib + did 



but this is the product of 



these determinants are respectively the determinants of T and Ti, 

 the components of T2. Hence the determinant of a transformation, 

 T2, which is the resultant of the transformations T and Ti, is equal to 

 the product of the determinants of T and Ti. 



This result is capable of immediate extension ; for let T a , Tb and 

 T c denote three transformations, the result of whose successive applica- 

 tions is equivalent to Td ; the compounding of T a and Tb is equivalent 

 to a third transformation, Tab. The resultant of Tab and T c is Ta, and 

 the determinant of Td is equal to the product of the determinants of 

 Tab and T c ; hence the determinant of Td is equal to the product of 

 those of T a , Tb, and T c . This mode of reasoning is applicable to the 

 resultant of any number of transformations ; hence by induction we 

 infer the following theorem : 



Theorem 1. The resultant T n of n projective transformations Ti 

 (i = 0, 1, 2, ... . n-1 ) is a projective transformation, and the deter- 

 minant of the resultant is equal to the product of the determinants of 

 the components. 



Inverse and identical transformations. — The transformation T 

 expressed by 



Xi 



ax + b 

 ex + d 



