K. Pearson 
291 
Using (ix) : 
/i2= 5-449,637, 
fi^ = - 3-472,590, 
90-747,442. 
The divergence between these results and those given by Mr Sheppard's 
process is very small and solely due to the arithmetic being cut off at the sixth 
place of decimals in the multiplication. Thus /ia' above really ends with G, and 
this difference is sensible in the fourth place of decimals of yUj when we multiply 
/ia' by 6/Ai'2. 
Now let us see if Mr Sheppard's process is in this case justified ; let us no 
longer put the as zero, i.e. no longer suppose high contact at the high fecundity 
end of the curve. We have 
Hi = 19/2000, 712 = 49/2000, «3 = 127/2000, = 204/2000. 
Hence we find from (xv) 
_ 035,729, 73 = -080,344, 7^ = - -136,750, 75 = -080,625. 
This leads us by (xvi) to 
^/ = V,' - -000,1089, n.: = v.; - + ■000,2525, 
^3' = -\v^ + -000,1510, = v: -\ v.; + - "000,1886, 
or the /i."s are only influenced in the fourth place of decimals. Substituting the 
values of v^, v^, v-( and about one end of the range just found, we have 
/*/= -494,391, yu;= 5-694,4195, 
/a/ = 4-733,026, /i/ = 91-933,7284, 
which lead by (ix) to 5-449,997, 
^3 = - 3-471,085, 
^l,= 90-745,703. 
We see that modifications are in the third place of decimals of /u.3 and fx^ and in the 
fourth of 1X.2. Clearly we are not justified theoretically in assuming high contact 
at the high fecundity end of the frequency curve, but for most practical problems 
Sheppard's corrections would in this case have been quite sufficient. The actual 
slope of the tangent to the frequency curve at the end of the range would be 
-g = y.= 035,729, 
which is of course fairly small. Thus if there be not high contact at one end of 
the curve, but the slope of the tangent be not large, Sheppard's corrections will 
still give the substantial part of the required correction. If, however, as in the 
mortality due to various diseases the curve meets the axis at a considerable angle, 
we must endeavour to determine in some such maimer as the above the value of 
the corrective terms. 
29—2 
