332
ON THE CENTRO-SURFACE OF AN ELLIPSOID.
[520
The Nodal Curve. Art. Nos. 29 to 60.
29. If two different points on the ellipsoid correspond to the same point on the
centro-surface, this will be a point on the Nodal Curve: the conditions for this if
(P v)> (p> Vi) are the parameters for the two points on tho ellipsoid, are obviously
(a 2 4- |) 3 (a 2 + rj) = (a 2 4- p) 3 (a 2 4- Vi),
(b 2 + p) 3 (b 2 +v) = + P ) 3 (b 2 + Vi)>
(c 2 4- |) 3 (c 2 + v) = (c 2 + p) 3 (c 2 + 77^;
these equations in effect determine 77 as a function of p so that the equations
— ¡3ya 2 x 2 = (a 2 4- p) 3 (a 2 + 77), &c.
then determine the coordinates (x, y, z) of a point on the Nodal Curve in terms of
the single parameter p
The relation between £ and 77 would be obtained by eliminating p, 77 : from the
foregoing equation: but it is easier to eliminate 77 and thus obtaining between
P and £ a relation in virtue of which ¿5 may be regarded as a known function of
£; 77 and 77j can then be expressed in terms of p ¿5, so that each of these quantities
will be in effect a known function of £( x ).
30. The relation between p p is in the first instance given in the form
a 2 [(a 2 4-1) 3 — (a 2 4- p) 3 ], (a 2 + £) 3 ,
b 2 [(6 2 + P 3 - (b 2 + p) 3 ], (V + Z) 3 ,
c 2 [(c 2 4- f) 3 - (c 2 4-10 3 ], (c 2 4- p) 3 ,
(a 2 + P) 3
(6 2 + P) 3
= 0.
(c 2 4- P) 3
Throwing out a factor (f—p) 2 , this becomes
2 [a? [3a 4 4- 3a 2 (f 4- p) 4- p 4- f p 4- p 2 }
X(6»-C ! ).(l, 1, !]№ + f)(# + (,), (i> s + f,)(c 5 + ?)>] = <>,
where the left-hand side is a symmetrical function of p p vanishing for f = p, and
therefore divisible by (£ — p) 2 ; it is also divisible by A, = (Z) 2 — c 2 ) (c 2 — a 2 ) (a 2 — b 2 ) (= a/3y).
To work this out, write f 4- P = y>, £p = the equation may be written
2 {(6 2 — c 2 ) a 2
3 a 4
4- 3a 2 /)
4-_/) 2 - #
3 Z> 4 c 4
4- 36 2 c 2 (6 2 4- c 2 )/)
4- (Z> 4 4- c 4 ) (p 2 - 5»)
4- 6 2 c 2 (/) 2 4- 8</)
4- 3 (b 2 + c 2 )pq
4- 3 q 2
where the left-hand side divides by A (p 2 — 4q).
1 = 0,
1 This was my first method of solution; and I have thought the results quite interesting enough to
retain them—but it will appear in the sequel that I have succeeded in expressing £, 17, £ 1} -q x , in terms of a
single parameter <r.
