958] 
ON THE SURFACE OF THE ORDER Yl. 
535 
positive, and hence x k — y p -z v ic p is a duad having the property in question, or changing 
the notation say x a — \fziw* has the property in question; and in like manner, by 
considering the several cases that may happen, we have to show that each of the duads 
x a — y p zvu) s , — x a zytu 8 , zy — x a yho 8 , w s — x a y&zy, 
x a yP — z y w 8 , x a z y — yPiv s , x a w 8 — y^zy, 
has the property in question; it being of course understood that, in each of these 
duads, the two terms have the same degree and the same weight. The first form 
cannot exist; for we must have therein a = /3 + y 4- 8 and 0 = /3 + 2y + 33, which is 
inconsistent with a, /3, 7, 8 each of them positive. For the second form /3 = a + y + 8, 
j3 = 2y + 38: this is a = 7 + 28 or the duad is y 2 r +3S — xy +28 zyw 8 , = (y 2 )? y 38 — (xz)y (x 2 w) 8 . 
Writing y 2 = xz — R, we have terms containing the factor R, and a residual term 
(xz)y {y 38 — (xhv) 8 }, and writing herein 
xw = yz — Q or xhv = xyz — Q, 
we have terms containing Q as a factor and a residual term 
(xz)y {y 3S - (xyz) 8 }, = (xz)y y 5 {(y 2 ) 5 - (xz) 8 }, 
and again writing herein y 2 = xz — R, we see that this term contains the factor R: 
hence the duad in question consists of terms having the factor R or the factor Q. 
Similarly for the other cases: either a, /3, 7, 8 can be expressed as positive numbers, 
and then the duad consists of terms each divisible by P, Q, or R; or else a, /3, 7, 8 
cannot be expressed as positive numbers, and then the duad does not exist: thus for 
the third form zy - x a y^w 8 , here 7 = a + ¡3 + 8, 2y = /3+38, or say 7 = 3a + 2/3, 8 = 2a + /3, 
and the duad is z' Sa+2f3 — x a yhv 2a+ P, = z 3a (z‘ 2 ) 2fi - (xw 2 ) a (ytvf, which can be reduced to 
the required form. But for the duad x a y< i — zyw 8 , we have a. + ¡3 = 7 + 8, /3 = 2y + 38, 
which cannot be satisfied by positive values of a, /3, 7, 8, and thus the duad does 
not exist. 
A surface of the order n which passes through 3n +1 points of a cubic curve 
contains the curve: hence the number of constants, or say the capacity of a surface 
of the order n, through the curve P = 0, Q = 0, R = 0, is 
£ (n -I-1) (n + 2) (n + 3) — 1 — (3n + 1), = £ (n 3 + 6?i 2 — 711 — 6). 
Primd facie the capacity of the surface AP + BQ + GR— 0, A, B, C being general 
functions of the order n — 2, is 
3 . £ (n — 1) n(n + 1) — 1, = \ (n 3 — n — 2), 
but there is a reduction on account of the identical equations 
xP +yQ + zR = 0, yP + zQ + wR = 0, 
which connect the functions P, Q, R : for n = 2, the formulae give each of them as 
it should do, Capacity = 2; viz. the quadric surface through the curve is