520] 
ON THE CENTRO-SURFACE OF AN ELLTPSOID. 
363 
Reciprocal Surface. Art. No. 86. 
86. The centro-surface is the envelope of 
a?x 2 by 2 c 2 z“ 
(a? + Jf + W+Jf 1 (c 1 + if 
1=0 
hence the reciprocal surface in regard to the sphere x 2 + y* + z 2 — № = 0 is the 
envelope of 
that is, 
(—jyX 2 + F 2 + # - £ 2 = 0, 
a 2 b- 
& 
arX- + b-Y' 1 + c 2 Z- — k- + 2£ {X 1 + F 2 + Z 2 ) + £ 2 (— + 77 + 
X 2 F 2 X 
c 2 
0, 
viz. the envelope is 
(a 2 X 2 + b 2 Y 2 + cX 2 - ¥) (—■ + - (X 2 + F 2 + X) 2 = 0, 
or, expanding and multiplying by arb' 2 c 2 , this is 
a 2 (6 2 - c 2 ) 2 FX 2 + 6 2 (c 2 - aj XX 2 + c 2 (a 2 - 6 2 ) 2 X 2 F 2 
— /c 4 (be 2 X 2 + c 2 a 2 F 2 + a 2 b 2 Z*) = 0, 
or, what is the same thing, 
a 2 a 2 FX 2 + b‘ 2 l3 2 Z 2 X‘ 2 + c 2 y 2 X 2 F 2 — lc 4 (6 2 c 2 X 2 + c 2 a 2 F 2 4- a 2 6 2 Z 2 ) = 0, 
which may be written 
a 2 FX 2 + bX 2 X 2 + c 2 X 2 F 2 + f 2 X 2 + g 2 F 2 + hX 2 = 0, 
where 
(a, b, c, f', g, h) = (aa, b/3, cy, 2k' 2 bc, 2k 2 ca, 2k-ab), 
and consequently, 
af + bg + ch = 2Ic 2 abc (a + /3 + 7) = 0. 
It would doubtless be interesting to discuss this surface as it here presents itself, 
and with reference to its geometrical signification as the locus of the pole, in regard 
to the sphere, of the plane through two intersecting consecutive normals of the 
ellipsoid : but I abstain from any consideration of the question. 
Delineation of the centro-surface for given numerical values of the semiaxes. 
Art. Nos. 87 and 88. 
87. I constructed on a large scale a drawing of the centro-surface for the values 
a 2 = 50, b- = 25, c 2 = 15. 
(These were chosen so that a, b, c should have approximately the integer values 7, 5, 4, 
and that a 2 + c 2 should be well greater than 26 2 ; they give a good form of surface, 
46—2