ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 107 
(ii6.) For the complete normal figure, the mean square of deviation from the 
mean is ; 
(hi,) If X* denote the mean ^th power of the deviation from the mean, 
K+2 — (^' + 1 ) — (^’ + 1 ) KKi 
which is the statistical equation to the curve. 
This equation gives 
X,,_i = 0 
X., =: {2s - 1) (25 - 3) ... 1 . X’, = 
> 
The Surface of Revolution of the Normal Curve. 
§ 6. Projective Solids and Surfaces. —Let S be a surface whose equation referred 
to three rectangular axes OX, ^OY, OZ, is of the form z = (f) (x). cf) {y). Then if 
we take sections of N by a system of planes parallel to OZX, and project these 
sections on OZX, we obtain a system of curves which are the orthogonal projections 
of one another with regard to their common base OX. Similarly if we take sections 
by planes parallel to OZY. On this account it is convenient to call such a surface 
a projective surface. If the surface is terminated in all directions by the base- 
plane OXY, the volume included between this plane and the surface will be called a 
projective solid. 
For the geometrical definition of a projective solid it is sufficient that the solid 
should be bounded by a plane base OXY, and that two lines OX, OY in this plane, 
at right angles to one another and to a line OZ, should be related to the solid 
in such a way that the sections of the surface by planes parallel to OZX, when 
projected on OZX, form a system of curves in orthogonal projection. If this is the 
case, it follows at once, from the elementary properties of projection, that the same 
property holds for sections by planes parallel to OZY. 
The sections of the solid by the two sets of planes parallel to OZX and to OZY will 
be called prineipal sections. 
The following properties of a projective solid are easily obtained from the 
geometrical definition. 
(i.) Let WR and MP be any two ordinates, and let the other ordinates in which 
the principal sections through WR and MP intersect be NQ and nq. Then 
WR.MP = NQ.my. 
(ii.) In one of the principal sections through an ordinate WR, take any two ordinates 
NQ and N”Q' ; and in the other take any two ordinates nq and u'q (fig. 3). Draw 
the principal sections through these ordinates, and let them enclose (with the base 
and the upper surface) a volume V. Then WR.V = area NQQ'N' X area nqq’n'. 
(iii.) From (ii.) it follows that if we fix a principal section S, and take vaiiable 
P 2 
