122 



SCIENCE. 



[N. S. Vol. VIII. No. 187. 



The case 7i=5 gives the group of 120 quad- 

 ratic transformations of the plane which is 

 the subject of the present study. This group 

 contains a linear subgroup of order 244 ! 

 which permutes in all possible ways the four 

 fundamental points of the Cremona group. 

 This subgroup is isomorphic with Klein's 

 linear group of order 4 ! for which Professor 

 Moore has shown a division of the plane to 

 be given by a certain complete quadrangle 

 (including its diagonals) whose vertices are 

 the four points permuted. 



I. This affords a means of finding a geo- 

 metric representation for our quadratic group 

 as follows : 



(1) A linear fractional transformation is 

 found which throws the complete quad- 

 rangle for the Klein group into another 

 whose vertices are the four fundamental 

 points of the quadratic group, and which, 

 therefore, gives the division of the plane 

 for the linear subgroup. (2) The linear sub- 

 group is transformed by all the quadratic 

 operators of G120, giving four quadratic 

 subgroups conjugate with the linear sub- 

 group. (3) The division of the plane for 

 these quadratic subgroups differs from that 

 of the linear subgroup only by replacing, 

 each time, the three diagonal lines by cer- 

 tain three conies. (4) The division of the 

 plane for the main group is then given by a 

 composite of the five pictures belonging to 

 these five conjugate subgroups, and con- 

 sists of the original complete quadrangle 

 together with its diagonals and twelve con- 

 ies. A further study of the various sub- 

 groups shows the following conjugate sys- 

 tems of special lines or points : (1) A sys- 

 tem of ten elements consisting of the six 

 sides of the original quadrangle, which are 

 fundamental lines, and the four pencils of 

 ' directions ' at the four fundamental points. 

 (2) A system of fifteen lines consisting of 

 the three diagonals and twelve conies. (3) 

 A system of fifteen lines consisting of cer- 

 tain three conies not in the configuration 



and twelve ' direction ' lines through the 

 fundamental points. (4) Twelve real points 

 at each of which five lines intersect. (5) 

 Fifteen real points where four lines inter- 

 sect. (6) Twenty imaginary points of 

 three-fold intersection. (7) Thirty real 

 points of two-fold intersection. (8) Twenty 

 imaginary points lying by pairs on the six 

 sides and four pencils. (9) Thirty imagin- 

 ary points lying by pairs on the three diag- 

 onals and twelve conies. 



II. The Klein linear group of order 4 ! 

 also affords the means of finding the in- 

 variants of the quadratic group, as follows :. 

 (1) The complete form-system of the linear 

 subgroup comes from the known system 

 for the Klein group by the same transfor- 

 mation which throws the generators of the 

 former group to those of the latter. (2) 

 The most general invariant form of any 

 given degree under the linear subgroup is 

 then set up with arbitrary coefiBcients and 

 operated upon by the quadratic generator 

 which extends the linear subgroup to the 

 main group. (3) This doubles the degree 

 of the given form, and hence the only pos- 

 sibility for the existence of an invariant 

 under the quadratic grouxD is to so deter- 

 mine the arbitrary constants that a factor 

 in the variables may divide out, leaving 

 the original form. (4) Hence an invariant 

 form under a quadratic operator must be a 

 rational fraction, such that a common factor 

 in the variables will cancel from numerator 

 and denominator, leaving the original frac- 

 tion. (5) It is found that the most general 

 forms suitable for numerator and denomi- 

 nator of invariant fractions of the 6th, 12th 

 and 18th degrees respectively are : 



VI A, m^A^ + m^P'', m^A^ + m^AP^ + m^C 

 when the m's are arbitrary constants and 

 A = 2fq' — 6(pV + 5=') + 1^'pqr — 9r' 

 P^ =/9V' - 4(5V' + pV) + lSpqr^ — 21r' 

 C = 100/9V'-1242)H560/5V-2150/3V* 

 + 2826pgr'— 286(py5V +/5V) 



