36 



Proceedings of the Royal Irish Academy. 



«i> "2; ciS) "4 ; "s, "s; «7j as- We shall refer to these eight points 

 as the R points. 



3. — Let z = Ix + my be the equation of any line, and let us seek 

 the equation which determines the parameters of the five points of 

 section. Substituting for %, its value Ix + my in the equation of 

 the curve, we obtain 



A {Ix + myy - 2B {Ix + my) + C = , 



• (1) 



and consequently the five ^s of the points of section of the curve 

 with the line, which we shall denote by 61, Oo, 63, 64,, 9^, are 

 found from the equation 



A {16 + mf - 2jB {Id + m) + C ^ cf>{6) ^ 



(2) 



If we now investigate the change in 9 due to changes dl and dm 

 in I and m, we obtain, by differentiating the above equation, 



dcf>{9) 

 dO 



Now, 



d9 + 2{A{W+m) - £]{9dl+dm) = 

 A{l9 + vi)-^= JI7 



and ,^ = M<f>'{9), where Jf is a function of I and m; lience 



d9 



d9i {9i) dl + dm) 

 ^=4- 2 — = 0, 



Ji?i 



M4>'{9,) 

 9i being one of the five roots of the equation 



«^(^) = 0, and <^' {9,) ^ {9,- 9,) {9,- 9,) {9,- 9,) {9,- 9,) 



We now write 



d9i (9idl + dm) 

 2-?^ + 2S _ = 0, 



(3) 



9,d9i 



9,-d9i {9i^dl + 9,^dm) 



JH, Mcf>'{9,) 



M<1>'{9{) 



{9^'dl+9,dm) 



22^ = 



M<}>'{9,) 



' (4) 



where 2 denotes summation from 9^ to 9^ 



