j. F. Tocher 
321 
Since S (ei-6.2) = S (e^e.2-) = 0 on the assumption of normal correlation, and since 
S{€i'e2-) = nai^a.fil + 2?-,o-) on tlie same hypothesis, we have : 
■ii'S.'^p^ = Di^-Di.f [rivf + nv.? + 2nry,i\iu + nv-cv.? (1 + r^2)\ 
The term nvihi? {\ does not appreciably affect the result and may be 
neglected. Thus 
^Pa ^ '"■l"'2 [^^1^ + '''2" + 2ri2t>iV.,]^ (2). 
An inspection of the relative local differences of pa, in Table IX., reveals the 
fact that the entire west of Scotland is large sectioned and that the east country- 
south of the Forth is small sectioned, compared with the general population. 
These deductions can, of course, be made from the lengths and breadths them- 
selves, but the facts do not come out so clearly. It is seen that Argyll and Ayr 
have large cross sections because of their long-headedness, while in the Inverness 
group's large section is due to the group's broad-headedness. It is also to be 
noted that both Edinburgh and Glasgow have small product means. 
Diametrical Product. (See Maps I. and II.) This product L x B x H = 
is a useful character, supplying as it does, an estimate of the mean relative size 
of head in each of the asylum groups. The means and standard deviations were 
calculated from the following formulae, derived by the same process of reasoning 
and approximation as that employed to deduce the mean and standard deviation of 
the LB product. 
^ S (xiW^Xg) = miiiujjia [1 + r^.v^Vo + r-^nViV^ + ro-^v^Vx] (3) 
m{)n.^yi. [v-c + v.? + v.? + 1i\,v{V.2 + '^.r^v^v-i + 2?'.,;jU,V3]- (4). 
In the region of Scotland south of the Grampians and north of the Border 
and Galloway (that is, practically the Scottish Midlands), with the exceptions of 
the large towns of Glasgow, Dundee, Aberdeen and Greenock, the asylum inmates 
are large-headed or macrocephalic. In the above mentioned towns they are small- 
headed or microcephalic. Edinburgh and the rest of the country approximates 
to the mean diametral product. The large-headed or macrocephalic group consists 
of Argyll and Lanark (which are dolichocephalic), and Montrose, Perth and Stirling 
(which are brachycephalic). It is plainly evident that, excluding Edinburgh, the 
inmates of asylums in the great cities are smaller headed than those of the rest of 
the country, that is, tliey are a microcephalic population. The only significantly 
small-headed or microcephalic rural group is that of Elgin, where there are only 
71 male and 89 female inmates. Aberdeen females are medium sized or meso- 
cephalic, while Inverness females are smaller sized than the males. The distribution 
of relative size of head is shown as follows : — 
