Continuous Groups of Projective Transfor- 

 mations Treated Synthetically. 



BY H. B. NEWSON. 



Part II. 



In Vol. IV, No. 2 of this Quarterly the writer gave a tolerably 

 complete synthetic theory of the Projective Groups in one dimension 

 and promised a similar theory for the same groups in two dimen- 

 sions. The present paper is a partial fulfillment of that promise. 



^1. Construction of Projective Transformations. 



Let TT and tt^ be any two planes intersecting in the line 1 and 

 making any convenient angle with one another. It is known from 

 the theory of projection that when we have given any four elements, 

 lines or points, of the plane tt and the four corresponding elements 

 of TT^ then the projection of tt on tt^ is completely determined. 

 These four elements must be taken in perfectly general positions; 

 i. e. if four points be taken, no three can lie on a right line; if four 

 lines be taken, no three can pass through a point.* 



Let us suppose that we have given four lines a, /3, y, 8, in the 

 plane tt and the four corresponding lines a'^ , (3^ , yS 8^ in tt^ The 

 four lines a, (3, y, 8, intersect in the six points P, Q, R, S, T, U ; 

 the corresponding points in tt^ are P^, Q^, R^ 5% T^ U^ Let 

 us call the points in which a, (3, y, 8 cut the line 1, A, B, C, D. 

 Then we have in the plane tt on the lines a, (3, y, 8, the four ranges 

 (PQRA), (TSPB), (QSUC), (RTUD). If we construct in the 

 plane tt^ on the lines a^, (3^, y\ 8^ the points A^, B^, C^ D^, so 

 that the anharmonic ratios (PiQiRi Ai)=(PQRA), (TiS'PiB^) = 

 (TSPB), (QiSiUiCi) = (QSUC), (RiTiUiDi) = (RTUD); then 

 A^, B^, C, Di will lie on the line P, which in tt^ corresponds to 1 

 in TT. Let the lines joining AA^, BB^, CCi, DDi bea', b', ci, d^; 

 the lines a^, b^, c\ d^ 1, P all lie in tt^ and touch a conic K', every 

 tangent to which cuts 1 and P in corresponding points. 



Let us now consider the line 1 to belong to the plane tt^ and find 

 the line 1^ in tt which corresponds to 1 in tt^. This can be done 



*See the following standard references: Reye's Geometrie der Lage, Band II, page 

 7. Clebsch-Lindemann's Vorlesungen ueber Geometrie, Band I, page 250. Lie's Con- 

 tinuierlicbe Griippen, page 21. 



(i43) KAN. UNIV. QUAR., VOL. IV, NO. 4. APRIL, lt<9U. 



