308 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(1) T'^A T=A, 



(2) TA T=A, 



(3) T-^ A T=A. 



The transformations of V constitute a group, as do also the transfor- 

 mations of r'" ; the transformations of r" that are commutative form a 

 group. 



In a paper entitled " Note on the Automorphic Linear Transforma- 

 tion of a Bilinear Form/' * I have shown that each transformation 

 T of the family T' belongs to a group with a single parameter of 

 transformations of F', and can thus be generated by an infinitesimal 

 transformation of T' ; and that a similar theorem holds for the family 

 r'". But on the other hand, for certain forms J, that T" contains 

 transformations (of determinant -t-1 as well as of determinant — 1) 

 that cannot be generated by an infinitesimal transformation of r".f 

 In this paper I consider only real forms J, and only real transfor- 

 mations of the variables x and y. Thus in what follows the matrix 

 A and the families T', T", V" of transformations T will be assumed to 

 be real. And with this restriction to real forms and families of real 

 transformations I show severally that each of the families T', T", T'", 

 for certain forms J, contains transformations (of positive as well as of 

 negative determinant) that cannot be generated by infinitesimal trans- 

 formations of that family. t A transformation of either of the families 

 r which is the second power of a transformation of that family I term 

 a transformation of the Jirst kind, otherwise a transformation of the 

 second kind ; and, for each of the families V, I show that every transfor- 

 mation of the first kind, but no transformation of the second kind, can 

 be generated by an infinitesimal transformation of that family. I also 

 show that a transformation of either of the families T is a transforma- 

 tion of the first kind when no negative number is a root of the charac- 



* These Proceedings, 31, 181. 



t If J" is a transformation of r", then \T\~ = \. Tliat r" shall contain trans- 

 formations of the type mentioned and of determinant +1, it is suflBcient that two 

 of the real roots of the characteristic equation of jF shall be of opposite sign and 

 equal in absolute value. 



\ In order that r', r", V" sliall severally contain transformations of the type 

 mentioned, it is sufBcient for r' that the real roots of the characteristic equation 

 of A shall not all be distinct; for T", that the ratio of two of the real roots of this 

 equation shall be equal to —1; and for T", that the ratio of two of the real 

 roots of the equation shall be negative. 



