66 REPORT—1899. 
We have : 
Re I 
0 a oP Or, 9) 
a 1 F 
a ems (iv.) 
where we write : F (r, v) = e-»" G (r, v). : ; ; : ane ie) 
It was shown in a preliminary report! that : 
¥F (7¢j9)= Hed (cos i) ee ae, OO : a 28) 
FE 
where tan 9=v and x (7, @) is a function which can be fairly easily calcu- 
lated, when certain preliminary functions have been tabulated. These 
X1» X39 Xs X7 functions were calculated in the preliminary report above 
referred to.” 
Now, in actual statistical application * may take as large a value as 
40 to 50. Hence, if cos be taken from the tables (cos ¢)"** is liable to 
a large error often reaching to the fifth place of figures when we are 
tabulating log F (7, v). Clearly, for accuracy, it is better not to find 
F (7, v) by interpolating between two tabular values of log F (7, v), but 
to deal with some new function in which (cos ¢)"*! does not occur, and 
then multiply by the actual value of (cos ¢)"*! deduced from the exact 
value of the angle ¢ and the quantity 7. This will not, of course, free us 
from the error, which arises from a value of cos ¢ tabulated to only 
seven figures being raised to a high power. The value of (cos »)’*? must, 
therefore, be found from 10-figure logarithmic tables of trigonometrical 
functions like those of Vega’s: ‘Thesaurus Logarithmorum Completus’ 
of 1794. But the error due to the determination of (cos )’*' from 
7-figure tables is not significant in the case of statistical investigations. 
For y), as determined for any observed frequency series—probably not 
containing more than 1,000 to 4,000 observations at a maximuin—is subject 
to a considerable percentage error.* It seemed, accordingly, desirable to 
tabulate for statistical purposes a function which is without the factor 
(cos p)"*?, and has yet a real statistical importance. This function is 
obtained in the following manner. The frequency y, per unit of variable 
v at the mean for the normal curve : 
—K(rJa)? 
Y¥=74) © ’ 
where o is the standard deviation, is given by 
jee 
: Oo ay 
where « is the area of the normal curve. For the curve (i) it is given* by 
ny! v =2x(1", #) 1 
‘ Up — ae ee BCS ho. ie ' ° vil. 
e a v/ 2 r—l ( ) 
1 B.A, Trans., Report 1896. 
* The following slip bas been since discovered in the tables of that report: 
loz x, for ¢ = 25°, should be 2.677,7543, and not 2.667,7543, as tabulated. 
3 See Phil. Trans. vol. exci. A, p. 297 et seg. ; numerically, perhaps, the error may 
amount in practice to *5 to 2 per cent. 
4 see Phil. Trans. vol. exci. A, p. 298 (equation cxxxvi,), where, however, the 
symbol x isused for 2 x (7, p) of the present notation and of that of the Prediminary 
Report. 
