ERROR TO OASES OF NORMAL DISTRIBUTION AND CORRELATION. 109 
Thus every vertical section of the solid is a normal figure of the same parameter, 
having its central ordinate in the plane through the axis at right angles to the plane 
of the section. 
It follows from § '6 that the solid is a projective solid, any two vertical sections at 
right angles to one another being regarded as principal sections. 
§ 9. Converse Propositions .—There are two converse propositions. 
(i.) If two principal sections of a projective solid are normal figures of equal para¬ 
meter, the solid is one of revolution. 
Let this parameter be 2a. From § 2 it follows that every principal section is a 
normal figure of parameter 2a. The solid will obviously have a maximum ordinate 
OH ; and each of the two principal sections through OH will contain the central 
ordinates of all sections by planes perpendicular to it. Take any other section 
through OH ; and let MP be any ordinate in this section. Draw planes through 
MP cutting the principal sections through OH in ordinates NQ and nq. Then the 
sections NQPM and ?igPM are normal figures of parameter 2a, having NQ and nq 
for their central ordinates. Let the tangents to these sections and to the section 
OHPM cut the respective bases in T, t, T' (fig. 5). Then PT, PT', P^ all lie in the 
tangent plane to the surface at P, and therefore TT'^ is a straight line. Also 
NM. MT — a~ — nM. M^, so that ON ; NM :: TM : Ml Hence the triangles ONM, TM^ 
are similar, and angle MTi = angle NOM ; and therefore a circle goes round NOTT'. 
Hence OM.MT' = NM.MT = a", and therefore the section OHPM is a normal figure 
of parameter 2a, having OH for its central ordinate. This is true for every section 
through OH, and therefore the solid is one of revolution. 
