14 
[709 
« 
709. 
ON THE NUMBER OF CONSTANTS IN THE EQUATION 
OF A SURFACE PS-QE = 0. 
[From the Tidsskrift for Mathematik, Ser. 4, t. iv. (1880), pp. 145—148.] 
The very important results contained in Mr H. Valentiner’s paper “ Nogle 
Seetninger om fuldstsendige Skjseringskurver mellem to Flader” may be considered 
from a somewhat different point of view, and established in a more simple manner, 
as follows *. 
Assuming throughout n > p + q, p > q, and moreover that P, Q, R, S denote 
functions of the coordinates (sc, y, z, w) of the orders p, q, n — q,n—p respectively: 
then the equation of a surface of the order n containing the curve of intersection of 
two surfaces of the orders p and q respectively, is 
so that the number of constants in the equation of a surface of the order n satisfying 
the condition in question is in fact the number of constants contained in an equation 
of the last-mentioned form. Writing for shortness 
a P = i (P + !) (P + 2 ) (P + 3 ) ~ 1, = iP (P* + + H), 
the number of constants contained in a function of the order p is = a p +1 ; or if 
we take one of the coefficients (for instance that of sc?) to be unity, then the number 
* Idet vi med stor Glasde optage Prof. Cayley’s simple Forklaring af den Reduktion af Konstanttallet i 
Ligningen PS—QR—0, som Hr. Valentiner havde paavist (Tidsskr. f. Math. 1879, S. 22), skulle vi dog 
bemerke, at Gründen til, at dennes Bevis er bleven saa vanskeligt, er den, at ban tillige har villet bevise, 
at der ikke finder nogen yderligere Reduktion Sted.