: THEORY OF STATISTICS. 
strictly normal, but, as a fact, a rough test suggests that they 
might have done so. 
10. Next we note that the distributions of all arrays of a 
normal surface should themselves be normal. Owing, however, 
to the small numbers of observations in any array, the distributions 
of arrays are very irregular, and their normality cannot be tested 
in any very satisfactory way: we can only say that they do not 
exhibit any marked or regular asymmetry. But we can test the 
allied property of a normal correlation-table, viz. that the totals 
of arrays must give a normal distribution even if the arrays be 
taken diagonally across the surface, and not parallel to either 
axis of measurement (cf. § 6). From an ordinary correlation- 
table we cannot find the totals of such diagonal arrays exactly, 
but the totals of arrays at an angle of 45° will be given with 
sufficient accuracy for our present purpose by the totals of lines 
of diagonally adjacent compartments. Referring again to Table 
III, Chap. IX., and forming the totals of such diagonals (running 
up from left to right), we find, starting at the top left-hand 
corner of the table, the following distribution :— 
0-25 78°75 
2 81-25 
3.25 665 
6-25 5925 
8 42-25 
9-75 30-75 
17 29-25 
345 19 
42 10°75 
46°25 1 
605 4-25 
67-5 3:5 
85°75 1-75 
87:25 1 
78 0-25 
94-25 — 
Total 1078 
The mean of this distribution is at 0359 of an interval above the 
centre of the interval with frequency 78: its standard-deviation 
is 4757 intervals, or, remembering that the interval is 1/,/2 of 
an inch, 3:364 inches. (This value may be checked directly from 
the constants for the table given in Chap. IX., Question 3, p. 189, 
for we have from the first of the transformation equations (8), 
03 =o07. cos’ 0+ a3 sin’ 0 + 2ry,00,. sin 6 cos 0, 
294.