PART II* 
FINITE GROUPS OF LINEAR HOMOGENEOUS TRANS 
FORMATIONS 
CHAPTER IX 
PRELIMINARY THEOREMS 
Linear Transformations, §§ 75-82 
75. Introduction and Definition. It is often of importance 
in analysis to exchange one set of variables for another, the 
variables of either set being linear homogeneous functions of 
the variables of the other set (cf. Ch. XVIII), as in coordinate 
geometry: 
x=x' cos 9—y' sin 6, 
(1) 
y=x' sin 6+y' cos 9. 
We assume that a function f(x, y) is given, in which the new 
variables (xy') are to be put in place of the old (a, y) by means 
of (1); this is called operating upon f by the linear transforma 
tion (1). 
A capital letter is in general used to denote a linear trans 
formation; thus, we shall here denote (l) by S. The result 
of operating upon f{x, y) by S may then be indicated sym 
bolically as follows: 
(2) (J)S=f{x' cos 9—y' sin 9, x' sin 9+y' cos 9). 
* This part was written by H. F. Blichfeldt. 
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