122 Proceedings of the Royal Society of Edinburgh. [Sess. 
For exact divisibility the coefficients of the various powers of x in the expression so 
obtained must be equal. 
The test, then, for finding whether an expression can be obtained by performing 
the operation \n\ on a simpler expression, is to arrange the coefficients of the various 
terms of the given expression in order in n columns ; the sum of the members in 
each column is taken, and if these sums are equal, the expression can be obtained 
by summation. 
It may be pointed out that these sums give a clue to other summatory factors which 
the expression may have, e.g. in the case worked out we observe that 70 has as its 
factors 2, 5, 7. Hence it is probable that the expression may have been obtained 
by performing the operation [2], [7], and a repeated [5] on a simpler expression. 
Before testing for the second [5] it is necessary to eliminate the first by a process 
of division. It will be found that Spencer’s formula is not obtained by an operation 
[2], but is obtained by an operation [7] on a simpler expression. The sums of the 
members in the various columns contain all remaining summatory factors, but they 
may have factors which do not belong to the summatory process. 
§ 6. Tests Performed on Actual Data. 
We shall now consider the relative merits of summation formulae and 
least-square formulae as tested by their performance when applied to 
definite numerical data. 
We have first to decide what is to be accepted as the measure of good 
performance in a graduation. Some previous writers have taken the 
criterion of goodness to be the smallness of the third differences of the 
graduated values. It is evident however that this, taken by itself, is 
worthless as a criterion, since it takes no account of the magnitude of the 
differences between the graduated and ungraduated values — in other words, 
it takes no account of the sacrifices that have had to be made in departing 
from the original data. We can, in fact, quite easily devise a graduation 
formula which will make the third differences of the graduated values as 
small as we please, or which will even make all of them absolutely zero, 
provided we are willing to sanction sufficiently large differences between 
the graduated and ungraduated values. Such a formula would come 
out first in order of merit when judged by the criterion of small third 
differences, although its absurdity would be obvious. If we are to have a 
criterion in any way resembling this, it would seem best to adopt that 
which has been proposed by Professor Whittaker, and which may be stated 
thus : Let u s ' be the graduated value corresponding to the ungraduated 
value u s : then the merit of a graduation is to be estimated by the small- 
ness of 2(u s ' — u s ) 2 + X2( A 3 u/) 2 , where X is a constant whose magnitude 
depends on the weight of the observations, i.e. X measures the extent 
