﻿vom 29. April 1878. 311 



in like manner the equation T = takes the form 



A'x 3 -+- zB'?? + 3 Ca + D' = , 



where J.', 5', C, D' are given functions of the coordinates £$ ! *jy£y«$ 

 and we have 



(47)' — £'G") 2 - 4 (4'C" — -ß' 2 ) (B'W — C n ) = , 



as the equation of the reciprocal surface; or (what is the same 

 thing) as that of the original surface, regarding J ,*?,£, w as 

 plane -coordinates. 



In regard to the foregoing equation T = 0, it is to be notic- 

 ed that if «i , #i , c x , /j , ^ , h x ; a 2 , b 2 , etc., a 3 ,b 3 , etc., instead of being 

 arbitrary coefficients, were the coordinates of three given lines 

 L x , L 2 , L 3 respectively; that is if we had 



«i/i + &i^i-r- Mi = ' , 

 «2/2+ hg-2 + ^2^2 = , . 



«3/3 H- hg 3 H- C3Ä3 = , 



then the three linear relations satisfied by (a , & , c , / , g , Ä) would 

 express that the line L was a line meeting each of the three giv- 

 en lines L x , L 2 ,h 3 : the locus is therefore the quadric surface 

 which passes through these three lines; and I have in my paper „On 

 the six coordinates of a Line" Camb. Phil. Trans, t. XI (1869) 

 pp. 290-323 found the equation to be the foregoing equation T= 0. 

 But it is easy to see that the same equation subsists in the case 

 where the three equation s a x f x -f- b l g l -{- dhx = etc. are not satis- 

 fied. For the several coefficients being perfectly general, any one 

 of the three linear relations may be replaced by a linear combi- 

 nation of these equations; that is, in place of a x , b x , c x fi , g x , Ä 1? 

 we may write a[ , b[ , c[ f{,g[ , h[ , where a[ = ß 1 a 1 -+- Q 2 a 2 + ö 3 a 3 , 

 b[ = S x b x -f- 6 2 b 2 -+- S 3 b 3 , etc. ; and these factors ß x , 6 2 » ^3 ma y De con ~ 

 ceived to be such that the condition in question a[fl + b[g[ -\- c[h{ 

 = is satisfied. Similarly the second set of coefficients may be 

 replaced by a 2 , ft 2 , c 2 , f 2 , gi , h' 2 where a 2 = cp x a x -\- cp 2 a 2 -\-cp 3 a 3 , etc. 

 and the condition a 2 f 2 -\- b' 2 g' 2 -\- c' 2 h 2 — is satisfied: and the third 

 set by a 3 ,b 3 , c 3 , / 3 ' , g' 3 , ä 3 where a 3 = vf/^ -h ^ 2 a 2 -+- \i/ 3 «3 , fite, 

 and the condition a 3 / 3 -+- 6 3 # 3 -+- c 3 Ä 3 = is satisfied. We 



23* 



