123 
1919-20.] On a Class of Graduation Formulae. 
to which we are willing to modify the original data for the sake of 
obtaining smoothness. 
In the present paper, however, I have used a test of a somewhat 
different type, which so far as I know has not been applied before, but 
which seems to have some advantages. It may be described thus : 
Consider some known analytic function of x, such as log x, of which 
tables accurate to (say) 6 places are available. A 4-place table of this 
function may be prepared by omitting the last two digits (which will be 
called the tail) and “forcing,” i.e. increasing the last retained digit by 
unity when the omitted tail begins with one of the digits 5, 6, 7, 8, or 9. 
We can regard the values of log x given by the 4-place table as affected 
with “ errors,” namely, the errors which have been produced by omitting 
the tails. Let us now take a sequence of these 4-place values, and graduate 
them by the graduation formula which is to be tested ; the effect of the 
graduation should be to smooth out the “ errors ” and restore, to some 
extent at least, the more accurate values of the 6-place table. The success 
with which this is performed may be taken as a measure of the merit of 
the graduation formula : for it must be remembered that the true purpose 
of a graduation formula is precisely to reduce the magnitude of accidental 
errors. The advantage of using a known function, such as log x, for the 
test is that we can be certain that the errors (viz. the tails) are accidental, 
i.e. non-systematic. It may, I think, be objected to the work of some 
previous writers on the subject of graduation that they have not attended 
sufficiently to the all - important distinction between accidental and 
systematic errors, and have not adequately guarded themselves against 
achieving smoothness by what amounts to the introduction of new 
systematic errors into the graduated values. 
a. In Table II this method of testing is applied to the function 
u(cc) = 0 , 01x 3 + 0T3 ; the last two digits are omitted, and the forced numbers 
so obtained are taken as the ungraduated data. Spencer’s formula and the 
least-square formula k = 1, m=10 are applied, and the graduated values 
are calculated to the same number of digits as the true values. The merits 
of the graduated values are easily obtained by comparing columns 9, 10, and 
11 ; the result is that the sum of the squares of the residual errors is 1082 
when the least-square formula is used, and 1712 when Spencer’s formula 
is used. 
f3. In Table III the method of testing is applied to the function 
10 7 
--39,999*95. Spencer’s formula and the least-square formula k = 1, 
m=10 are used. The merits of the graduated values are obtained by 
