140 MR. W. F. SHEPPARD OH THE APPLICATION OF THE THEORY OF 
square unity about a mean value {(M — Mi)/& sin D] cos D. Hence the correlation- 
solid is a projective solid whose vertical sections by planes parallel to OX are normal 
figures of semi-parameter unity ; and since the values of (M — are distributed 
normally v/ith mean square unity, the sections by planes at right angles to OX are 
also normal figures of semi-parameter unity; be., the correlation-solid is the standard 
normal solid. 
(ii.) By taking vertical sections parallel to OY, we see that the values of 
(L — Li)/a are normally distributed, so that the values of L are normally 
distributed; and that in any class distinguished by a particular value of L the 
values of M are distributed nortnally with mean square Ir siir D about a mean 
value Ml -f- — cos D. (L — Ld. In other words, if the distribution of L is correlated 
a ' 
with that of M, the distribution of M is correlated with that of L. 
(iii.) Conversely, if the correlation-solid of two distriburions is the standard normal 
solid, the distributions are normal and normally correlated. 
(iv.) We have already seen (§ 14) that wdien the distributions of L and of M are 
normally correlated, the values of Ih mM are distributed normally. We might 
obtain this result directly by the method adopted at the beginning of § 13. In 
the base-plane draw the lines whose equations, referred to OX and OY as axes, are 
la sin D . X -j- mb sin D . y = ^i, and la, sin D . cr fi- mb sin D , y = £ 2 - Then the 
vertical planes through these lines will include between them the elements repre¬ 
senting individuals for w hich I (L — Lj) -f m (M — Ml) lies between and ^ 2 - Draw 
the central vertical plane at right angles to these planes, cutting the two sections 
in the ordinates WjHi and W 2 B 2 - Then the number of these individuals is pro¬ 
portional to the area W 1 R 1 II 2 W 2 , be., it is proportional to the area of the standard 
normal figure included between ordinates at distances ^i/{l~o? -h 'llmab cos D + mW}^ 
and cos D-j-from the median; and therefore the values of 
ZLi fi- 7nMi are distributed normally with mean square fW ‘llmab cos D -h 'nrb“ 
about the mean value ILi -j- wMi. 
(v.) Since (§ 25) the correlation-solid of the distributions of ZL + ?nM and of 
Z L -j- wbM is also the standard normal solid, it follows (see (iii.) above) that these two 
distributions are normally correlated. 
§ 27. Determination of Divergence by Double Median Classification. —The portion 
of the solid which lies on the positive side of each of the two planes OZY and OZX 
(OZ being the axis of the solid) represents all the individuals for wdiich L and M are 
greater than L, and Mj respectively; and the portion which lies on the negative side 
of OZY and the positive side of OZX represents those for which L is less than Li and 
M greater than Mj. Bat, since the solid is a solid of revolution, these volumes are in 
the ratio of tt — D ; D. Hence, if we arrange the whole number of individuals in 
four classes, thus :— 
