Kansas University Science Bulletin. 



Vol. I, No. 11. NOVEMBER, 1902. | ^xl no. 1 ??.' 



COLLINEATIONS OF SPACE WHICH LEAVE INVARIANT 



A QUADRIC SURFACE. 



BY HELEN B. BREWSTER. 



1. In an article published in the "Annals of Mathematics," second 

 series, Vol. II, No. 4. Dr. Ruth Gr. Wood has discussed " Collin eations 

 of Space which Transform a Non-degenerate Quadric Surface into 

 Itself." In his "Theorie der Transformationsgruppen," Vol. Ill, pp. 

 251-254, Lie has given a table of groups of transformations having 

 the above property, these groups having been developed analytically. 

 Prof. H. B. Newson has developed a theory of collineations of space, 

 his work being synthetical. 



The object of this paper is to follow the last-named method,' and 

 thereby determine a table of all groups of collineations of space 

 which leave invariant a non-degenerate quadric surface. This method 

 has been found to admit more thorough classification than that of 

 Doctor Wood. 



An acquaintance of the reader with Professor Newson's work has 

 been assumed, the language used being practically in complete agree- 

 ment with that of Newson's writings; but it has been found to be im- 

 practicable to attempt the same symbolic notation. No effort has 

 been made to repeat proofs given by Newson, and references to his 

 paper are frequent. My sincere thanks are due to Professor Newson 

 for the careful consideration he has given this work and the time he 

 has spent in the correction of errors. 



A close reading of the following table of symbols will obviate the 

 danger of vagueness in the language of the remaining work. The pre- 

 exponent and subscript of a given group denote, respectively, the type 

 and number of parameters of the group ; the invariant figure is de- 

 noted by the symbols enclosed in the parentheses, while constants of 

 the transformation are given as subscripts to the parentheses. Thus, 

 1 G3(lmF) a denotes a three-parameter group of type I, having for in- 



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