[339 
339] 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
193 
j order and 
The last formula, for the weight of the square + 2 system, was communicated to me 
by Dr Salmon, the others are all in effect given in the Appendix, “ On the Order of 
Systems of Equations,” to his Treatise on the Analytic Geometry of Three Dimensions; 
0, y,..\ and 
ises; and con- 
and in the investigation in the following Annexes 2 and 3, the route which I have 
followed was completely traced out for me by him, so that I have only supplied the 
details of the work. 
eeds by 1 the 
Annex No. 2.—Investigation of the formula for S (m 3 ), when the curve rn is the pq 
complete intersection, viz. when it is the intersection of two surfaces of the orders 
p and q respectively (referred to, Art. 40). 
Let U= 0, V= 0 be the equations of the two surfaces of the orders p and q 
respectively. Take (x, y, z, w) the coordinates of a point on the curve, so that for 
these coordinates we have U = 0, V = 0; and in the equations of the two curves 
respectively, write for the coordinates x + px', y + py', z + pz', w + pw'; then putting for 
shortness 
A = x'd x + y'dy + z'd z + w'd w , 
the resulting equations may be represented by 
(AU, A 2 U, ... ApUQI, py- 1 — 0, 
(AF, A 2 F, .. A«F$1, p)^- 1 = 0, 
where it is to be noticed that besides the expressed literal coefficients there are 
numerical coefficients (not as the notation usually denotes, the binomial coefficients, 
but ) =i-o>rb> &c - 
seds by 2 the 
Supposing that (x, y, z', w') are the current coordinates of a point on the line 
drawn through the point (x, y, z, w) to meet the curve in two other points, the 
equations in p must have two common roots, and this gives a system equivalent to 
two equations, or say a plexus of two equations. If from the plexus and the two 
equations U — 0, F = 0 we eliminate (x, y, z, w), we obtain an equation S' = 0 in 
(x', y', z', w), which is in fact the equation of the scroll S{m 3 ), taken (as is easily 
seen to be the case) thrice; that is, S (to 3 ) = l Degree of S'. But observing that the 
coordinates (x, y', z', w) enter into the plexus only and not into the functions U, V, 
and treating (x', y', z, w') as weight variables, Degree of S' — AVeight of System 
(Z7= 0, V = 0, Plexus) = Deg. U x Deg. V x Weight of Plexus, —pq x Weight of Plexus; 
or, writing pq = /3, 
+2A 2 A/-t-2aV. 
S (m 3 ) = |/3x Weight of Plexus. 
c. y. 25