MINDING’S THEOREM. 681 
This is a second cyclic cone, intersecting (11) in the four directions 8. Of course 
it is obvious that (11) and (13) are unaltered by the substitution of p+yf 
for p. 
If we look on B as given, while p is to be found, (11) is the equation of a 
right cylinder, and (13) that of a central surface of the second order. 
9. A curious transformation of these equations may be made by assuming p; 
to be any other point on one of the Minpine lines represented by (11) and 
(13). Introducing the factor —@?(=1) in the terms where 6 does not [{appear, 
and then putting throughout 
Bllai—p,  - . : : , (14) 
(11) becomes 
—p’pitS*ppi=S(pi—p)a(ri—p)  - + eS A) 
As this is symmetrical in p, pi, we should obtain only the same result by 
putting p; for p in (11), and substituting again for B as before. 
From (13) we obtain the corresponding symmetrical result 
(p°—Sppi) Spixpi + (P| —Spp:) Spap = —SppiS(pi—p)a(pi—p) —S(pi—p)@"(pi—p)_- (13’) 
These equations become very much simplified if we assume p and p, to lie 
respectively in any two conjugate planes; specially in the planes of the focal 
conics, so that Sé’p=0, and Sy‘p,=0. 
For if the planes be conjugate we have 
Spap, =0, 
Spa'pi=0, 
and if, besides, they be those of the focal conics, 
Sppi=—SP'pSh'p., 
Spo’p =eiSpap, &c., 
and the equations are 
—ppitS?ppi=SpiapitSpap, . . : (117) 
and 
p’Sprapit piSpap=—Spia'pi—Spw'p. - -  (18°") 
From these we have at once the equations of the two MINDING curves in a 
variety of different ways. Thus, for instance, let 
pi=po 
and eliminate p between the equations. We get the focal conic in the plane of 
