520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
357 
the term in [ ] is 
which is found to be 
= - 4o- 2 + <7 (8 + 15M + Sf. M 2 ) - 2M - pi 2 , 
and the whole numerator is thus 
3 (3a- — 2) (a -2 — a — ^M) [(3M + 4) er + M] 
- 4a 3 + a- 2 (8 + loM + $£ M 2 ) + a (- 2M- f M% 
which is 
= (36 + 27iH) a- 4 - (64 + 36i¥) a- 3 + 32o- 2 + 16M<r+ SM 2 . 
71. We have thus 
2 dx 7 (36 + 27i¥) o- 4 — (64 + 36.M) a s + 32a- 2 + 16Ma + SM 2 
= etct — 
a (a- — 1) (3a- — 2) (4a- + 3M) {(4 + 3i¥) a + M] ^o- + ^ 
and thence also 
2dz j (36 + 27M) a- 4 — (64 + 36M) a 3 + 32a- 2 + 16-Mo- + 3¥ 2 
— = der 7 
(j (a — 1) (3a- — 2) (4a- + 3M) {(4 + SM) a + M) ^a- — ^ ^ 
so that dx and dz also vanish when a- is a root of the quartic equation: the points 
in question are therefore cusps of the nodal curve. 
Centro-surface as the envelope of the quadric Xa 2 x 2 (a 2 + i;)~ 2 = 1. Art. Nos. 72 to 76. 
72. The equations — fiya 2 !*? = (a 2 + %) 3 (a 2 + rj), &c. considering therein £, vj as variable 
give the centro-surface; considering tj as a given constant but £ as variable they give 
the sequential centro-curve; and considering f as a given constant but rj as variable 
they give the concomitant centro-curve. 
73. Suppose first that rj is a given constant; to eliminate | we may write the 
equations in the form 
— (ßy) 3 iflaf ( a2 +v) s= ( a2 + £)> & c -> 
and then multiplying first by a. {a 2 + rj), &c. and adding, and secondly by a, &c., and 
adding (observing that Xa (a 2 + f) (a 2 + q) = — a/3y, Xa (a 2 + £) = 0); we have 
X (aax) 3 (a 2 + rj) :! = (aßy) 3 , 
X (aaxf (a 2 -f rj)~ 3 = 0, 
which equations, considering therein rj as a given constant, are the equations of a 
sequential centro-curve.