a) — a v + a 0 + ci n — p + ci n —
<'n—q
•v—a (¿n—p—q 1 •
^p—q
709] ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE.
15
of the remaining constants is = a p ; viz. a p is the number of constants in the equation
of a surface of the order p. As regards the surface in question
P, Q
E, S
= 0,
we may it is clear take P, Q, R each with a coefficient unity as above, but in the
remaining function S, the coefficient must remain arbitrary : the apparent number of
constants is thus = a p + a q + a n - p + CL n -q + 1 ; but there is a deduction from this number.
The equation may in fact be written in the form
I P + (*Q, Q
\ R + ctS + /3P + a @Q, S + (3Q
= 0,
where a. represents an arbitrary function of the order p — q, and /3 an arbitrary function
of the degree n—p — q: we thus introduce (a p _ q + 1) + (a n - p _ q + 1), = a p - q + an-p-q + 2,
constants, and by means of these we can impose the like number of arbitrary relations
upon the constants originally contained in the functions P, Q, R, S respectively (say
we can reduce to zero this number a p _ q + an-p-q + 2 of the original constants): hence
the real number of constants is
a p + a q + a n ~ p + a n _ q + 1 — (a p - q + a n - p - q + 2),
+ n — Cl't
p-q
CL n —
n-p-q
-1
— Up "p Qj q 4 U ¡y— p
= &) suppose ;
viz. this is the required number in the case n > p + q, p> q.
If however n=p + q, or p = q, or if these relations are both satisfied, then there is a
P', 0' i
further deduction of 1, 1, or 2: in fact, calling the last-mentioned determinant ^ ^ ,
then the four cases are
n>p + q, p>q,
n=p + q, p>q,
n > p + q, p = q,
n=p + q, p = q,
F, Q
R', 8'
F, Q'
R', S'
F, Q'
R', S'
F, Q'
R', S'
F, Q'
R', S'
P' + kR', Q'+kS'
R', S'
F, Q' + kP'
R', S' + kR'
P' + kR', Q'+ IP'+ kS'+ klR'
R', S'+ IR' |
where k, l denote arbitrary constants : these, like the constants of a and /3, may be
used to impose arbitrary relations upon the original constants of P, Q, R, S’, and
hence the number of constants is =<w, u> — 1, co — 1, a> — 2 in the four cases respectively,
where as above
