110 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
(il.) Tf a solid of revolution is also a projective solid, the generating figure is a 
normal figure. 
Let OH be the central ordinate. Then every vertical section is symmetrical about 
the plane through OH perpendicular to it, and any two vertical sections, if arranged 
so as to have their central ordinates coincident, will be interconvertible by projection. 
Let S be any section through OH, and let NQ and N'Q' be any two ordinates in 
this section, ON beine’ oTeater than ON'. Let the tancrents to S at Q and Q' cut 
ON' N in T and T'. 
Describe a circle in the base-plane on ON as diameter, and draw the chord 
NM = ON'. Draw the ordinate MP, and let the tangent at Q to the section MPQN 
cut MN produced in P (fig. 6). Then MP is the central ordinate of the section 
MPQN ; and therefore, since this section and the section OHQ'N' are interconvertible 
by projection, it follows that NR = N'T'. 
Since QR and QT are tangents to sections through NQ, QRT is the tangent plane 
at Q. The solid being a solid of revolution about OH, this tangent plane must 
be perpendicular to the plane OQT. The base-plane is also perpendicular to the 
plane OQT, and therefore TR, which is the line of intersection of the tangent plane 
and the base-plane, is perpendicular to the plane OQT. Hence OTR is a right angle, 
and therefore a circle goes round OMTR, so that ON . NT = MN . NR = ON'. N'T'. 
In other words, the rectangle ON . NT is constant for different positions of N, and 
therefore the central section is a normal figure. 
