1884.] 



Non-Euclidian Plane Geometry. 



and where the coefficients are connected by six equations only, the 

 system thus depending on three arbitrary parameters. If, as before, 

 we write P lf Q 1? P, Q, for iXj T l5 iXj + Yp *X Y, iX+Y respec- 

 tively, then the relation is readily found to be 



Q 



z 



Qi 



177 



17 + 7' 



this being read according to the lines only P 1 =i( + *'a' */ 



&c., not according to the columns : in order that the coefficients 



may be real, we must have , /3', 7', ft", 7", real, ft, 7, a', a" pure 



imaginaries. 



Writing the equations in the form 



Q 



PI 



Qi 



z, 



viz., P^AP + BQ + CZ, &c. 



it would be possible to deduce the equations which connect the new 

 coefficients ; but these are more easily obtained from the consideration 

 that we must have identically PjQj Z^^PQ Z 2 ; the equations are 

 thus found to be 



O, B" 2 -BB'=0, C" 2 -CC'=1 

 2A"B"-AB'-A'B=-1, 2A' / C"-AC'-A'C=0, 



2B"C // -BC'-B'C=0. 



18. The general theory of the transformation of a quadric function 

 into itself enables us to express the coefficients in terms of three 

 arbitrary parameters. There is no difficulty in working out the 

 formulae, and we finally obtain 



