P. F. EVERITT 
387 
differ, so that the checking not only applies to the actual numerical work but also 
tests the accuracy of the quadrature formula. The whole of the work was carried 
out using five places of decimals, and the resulting values of djN taken to five 
places gave a maximum discrepancy of one unit in the last place ; this discrepancy 
could always be traced to the effect of cutting off at the fifth place in the 
formation of the product zl and only occurred when, as must occasionally happen, 
a number of values grouped together by the quadrature formula happened to be 
rounded off in the same direction. 
In the special case h = k, the value of djN only occurs once in each table 
and these cases were checked by differences. 
Method of using the Tables. 
In using the Tables to find the value of r the quantities (b + d)jN, (c + d)jN 
and djN are first found by division and then the values of h and k taken out from 
Sheppard's Tables. Using these values of h and k as arguments and comparing 
the values of djN given in the Tables with the actual value of djN in the 
particular case, the two values of r, between which the required value probably 
lies, can be readily found by inspection. 
For each of these two values of r, the value of djN is next found by inter- 
polation, using the actual values of h and k as arguments ; for each djN there will 
be two such interpolations, one for h and one for k, and it is preferable as a 
general rule to perform that interpolation first for which the correction to djN 
is the smaller. Having now obtained the value of djN for the actual h and k 
for the two assumed values of r, the actual value of r is easily found by one more 
simple interpolation. The process may, at first sight, seem rather complicated 
but in actual practice the result is obtained with ease in a very few minutes. 
Example. 
Consider the fourfold table given below. 
608 
45 
653 
9 
48 
57 
617 
93 
710 
By division = -13099, = 08028, ^ = -0676, 
whence from Sheppard's Tables h = 11218, k = 1-4032. 
The values between which r will probably lie are next found by inspection of 
the tables, and it will at once be evident that the value of r lies between 
•9 and '95. 
49—2 
