108 
MR. W. P. SHEPPARD OH THE APPLICATION OF THE THEORY OF 
ordinates NQ and N'Q', the volume of the solid bounded by the other principal 
sections through NQ and N'Q' is proportional to the area NQQ'N'. 
(iv.) From (ii.)it also follows that if V is the whole volume of the solid, WR any 
ordinate, and A and A' the areas of the principal sections through WR, then 
WR.V = A.A'. 
Fig. 3. 
R 
(v.) Let OH be the ordinate passing through the centre of gravity of the solid, 
and let S and S' be the principal sections through OH. Tlien the central ordinates of 
all sections parallel to S (he., the ordinates through their respective centres of gravity) 
lie in S', and the central ordinates of all sections parallel to S' lie in S. 
§ 7. Normal Solid mid Normal Surface .—Let the half of a normal figure of 
parameter A'A = 2a, lying on one side of the central ordinate OH, be rotated 
about this ordinate through four right angles. The solid so formed will be called a 
normal solid, and its surface will be called a normcd surface. The plane traced out 
by the base will be called the hase-plane. A section of the solid by a plane 
perpendicular to the base-]jlane will be called a vertical section. 
§ 8, Normal Solid is Projective Solid .—Let S be any vertical section of the 
solid, and AIR any ordinate in this section. Draw ON perpendicular to the plane of 
the section, and let NQ he the ordinate at N. Let the tangents at P to the section 
S, and to the central section through MP [i.e., the section through AIP and the axis), 
cut the base-plane in T and T' respectively (fig. 4). 
Since PT and PT' are tangents to sections through P, the plane PTT' is the tangent 
plane to the solid at P. But the solid is a solid of revolution, and therefore this 
plane is perpendicular to the plane OMP. The base-plane is also perpendicidar to 
the plane OMP, and therefore the intersection TT' is perpendicular to this latter 
plane. Hence OT'T is a right angle, and therefore a circle goes round ONT'T, so 
that NM.MT = OM.MT'. 
But the section by the plane OMP is a normal figure of parameter 2a, and there¬ 
fore OM.AIT' = a?. Hence also NM.MT = a’; i.e., the section S is a normal figure 
of parameter 2a, having NQ for its central ordinate. 
