86 KEWSOK: LINEAR GEOMETRV. 



This readily reduces to 



(p^%wH)(D(/3- + a,wi>'3 -a,w2H) 

 k= — w :; -^ ] ; 



a 1) 

 where 1 ) 



a,h, 



I shall now gi\'e a geometric definition ut' an invariant of a binary 

 form. An invariant of one or more binary forms is any function of 

 the coefificients whose vanishing expresses the condition that the points 

 forming the group or groups have to hne another some relation unal- 

 tered by any projection whatever. 



The anharmonic ratio of four points is absolutely unaltered by any 

 projection whatever, and hence the above expression for k is an 

 absolute invariant of the system P and C. Moreover, by giving dif- 

 ferent values to k, we may obtain an infinite number of absolute 

 invariants of the system. Among this infinite number of invariants 

 are a limited number of ])rimary or fundamental invariants out of 

 which all the others may be compountled. We may obtain from the 

 above expression for k all these funilamental invaiiants by giving to k 

 its Critical \'alues. 



I must explain what I mean by critical values of k. The six 

 expressions for the anharmonic ratios of four points on a line are k, 



I I k — I k 



,1 k. , . and . A critical value of k is anv number 



k I k k k I 



w'hich substituted for k will cause two or more of the six expressions 

 to have equal values. Thus, i, o, and CO are critical values of k: and 

 the geometric meaning is, that two of the four points coincide. Also 



I 



— , — I, and 2 are critical values of k; and the four points then form 



2 



a harmonic range. Lastly, — w and — w^ are critical values of k, the 

 four points then forming an equi-anharmonic range. We have thus 

 eight critical values of k, and there are no more. 



These eight critical values of k substituted in the above equation 

 will not always give fundamental invariants; but all fundamental inva- 

 riants of the system will be included among those found by giving 

 these eight critical values to k; any compound ones thus obtained will 

 be easily recognized. 



We now proceed to give to k in the above equation the critical 

 values I, o, and oo. Substituting and reducing we obtain the follow- 

 ing results: 



2 1 2 



F'ork=x, we have p^---H=o; or Dp^ -a, (p^ — H):=o; 



