92 nf.wson: linear (;i:(1MEtrv. 



(.piartic, viz: 4a-'/ — alz-rj- o. In this eiiuation I ae — 4bd — 3c*, 

 and J^^ace-| 2bcd — ad- — eb"- — c-^ 



The anharmonic ratio of the four points represented by x Rjy=o, 

 X- R„y=o, X- R-jV— o, and x R^y=o is given by 



R, — R, R4 R, (1 p— 1 r)(, p-^ r) p— r z^-z. 



Rg— R3 Ro— R4 (1 r— J q)(— 1 q— 1 r)- q— r Zg— z. 



By solving the reducing cubic we find the following values of z,, z.,, 

 and Z.J : 



I \ 1 \" IV 



Zj U' I . /._, --\VU^- W' J , z.( w"-u^ — ^^'^f- 



u- 



Sa- 



Where v _, and u^^; i ~J^ \ 



Substituting these values of z,, /..., and /., in the above expressions 



for k we get 



\v u ' • w -" \' 

 k 



u^ . v 



The fundamental invariants of the quartic are now obtained from 



the above equation by giving to k its critical values. Let us give to k 



I 

 the values 1.2, 



W lien k I. we ha\e u'* — w-'v o; 



2 

 when k 2, we have u'^ -wv - o; 



1 ■-• 



when k— - , we have u'' - v o. 



2 



Kach of these expressions when transposed and cleared of radicals 

 leads to the same result. The ])roduct of the expressions is u^ — v^^^b, 

 which is free from radicals and imaginaries. Substituting the above 

 values of u and v in this equation, the expression reduces to J^o. 



] = ace— 2bcd — ad" — eb*— c^ is, therefore, a fundamental invariant 

 of the quartic; and is also the condition that the four points constitut- 

 ing the quartic should form a harmonic range. 



Giving to k its critical values w and - w^, we get the following 



results: 



when k^— — w, we have v o: 



when k^- — w^, we have v-o. 



I 

 Hut since v — , we have I— ae — 4bd--3c- as another funda- 



12a- 



mental invariant of the quartic. It is also the condition that the four 



points constituting the cjuartic should form an equi-anharmonic range. 



