341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
241 
Article Nos. 41 to 46.—Proof of identities for the fourth transformation. 
41. Consider the coefficients {a, h, c, f g, h) and the inverse set (Si, S3, (S, %, @, «£)), 
and the coefficients (a', V, c, /', g', h’), and the inverse set (21', S3', 6', ®, #') ; then 
we have identically 
(a , . y, z) 2 (81',. f£a, ..) — (81',. .\ax +hy + gz,. .) 2 
= (a', . .$>, y, zf (21, . fia',..) - (21,. .Ja'x + h'y + g'z,. .) 2 , 
where (21', . .$>,..) and (21, . .$a',..) stand for 
(21', 23', (S', g', ©', b, c, 2/, 2g, 2h) 
(SI, S3, <S, g, ©, «№', V, o', 2/', 2g', 2h') 
respectively. 
42. Taking (a, h, c, f, g, h), the second differential coefficients of a function If 
of the order to, and in like manner (a', b', c', f, g, li ), the second differential coefficients 
of a function U' of the order to', we have 
TO (TO -1) if • (21',. ffd x , dy, 0 2 ) 2 U' — (to — l) 2 (21',. f[d x U, dyU, d z uy 
= to' (to'-1) 0\(SI,..$a., d v> wu dyU', d z U') 2 ; 
and in particular if U' be the Hessian of U, then to' = 3to — 6. 
43. Hence writing 
o = (21, . .$0„ d y , 0 2 ) 2 H, V = (2Ï,. 0,#, d z H) 2 , 
Hi = (21',. .$9«, 0 ?y , 0 2 ) 2 TJ, dh = (21',. .$3.17, a^CT, 0 2 H) 2 , 
we have 
to (m — 1) Uilx — (to — l) 2 % = (3to — 6) (3m — 7) H£l — (3to — 7) 2 T ; 
or if Z7= 0, then 
— (to — l) 2 M r 1 = (3 in — 6) (3to — 7) HQ — (3 to — 7) 2 d r ; 
whence also 
— (to — l) 2 0d / 1 = (3to — 6) (3 m — 7) (HdQ + £ldH) — (3 to — 7) 2 0M/', 
which is the formula, ante No. 21. 
44. Recurring to the original formula, since this is an actual identity, we may 
operate on it with the differential symbol 0 on the three assumptions: 
1. (a, b, c,f, g, h), (21, 23, (S, g, «£> ) are alone variable. 
2. (a', b', o’, /', g', h'), (21', 23', (S', S', ©', £') are alone variable. 
3. (x, y, z) are alone variable. 
C. V. 
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