411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 349 
viz. the equations are read vertically downwards. The first, second, and third equations, 
and the sum of the fourth and fifth, all give the same relation, 132— SA — 7 = 0; 
there are consequently, inclusive of this, five independent relations. By combining the 
equations so as to simplify the numbers, I find these to be 
SA +1 -132 = 0, 
4>A-B-2G -80 = 0, 
7A -5 + 25 + 2F+ 2G- 476 = 0, 
26A-5 + 85 + 8 H -1532 =0, 
6 A — E — 25 + 12G-48#+ 40/ -132 = 0. 
56. I found, as presently mentioned, .4 = 110, 5 = 272, (7=44; values which satisfy 
(as they should do) the second equation; and then assuming 5 = 116 and 5 = 303, 
we have F = G = — ^-f 1 , H = — 248, /= — 198; and the formula is 
/3' = 2»(»-2)(llw-24) 
— (110?? — 272) b + 44g 
— (116n — 303) c + 
+ ^|iyS + 248 7 + 198i 
+ linear function (i, j, 6, x, C, B, j', 6', G, B'), 
the process not enabling the determination of the coefficients of the linear function. 
57. The values of A, B, G were found from the general theorem that if three 
surfaces of the orders /a, v, p respectively intersect in a curve of the order m and 
class r which is a-tuple on /a, /3-tuple on v, and 7-tuple on p, then the number of 
the points of intersection of the three surfaces is 
= p,vp — to (/3 7 p + yotv + <x(3p — 2a/3 7 ) + a/37?'. 
Apply this to the case of a surface of the order n with a nodal curve of the order b 
and class q, intersecting the Hessian and flecnodal surfaces, we have 
Order. Passing through (b, q), times 
Surface n 2 
Hessian 4?i — 8 4 
Flecnodal 11?a —24 11 
whence number of intersections is 
= 4w(w — 2)(lift —24) — 6 {n. 4.11 +(4w— 8) 11.2 + (ll?i — 24) 2.4 — 2.2.4.11} 
+ 2.4. llg, 
that is 
= 4?i (n - 2) (11?a - 24) - (220?? - 544) b-88q;