ON CALCULATION OF THE G (7, v)-INTEGRALS. 75 
@=20° 35’ are sensibly identical with those obtained by interpola- 
tion from a table of y’s proceeding by degrees. The tables of the 
x-functions will thus, till the F(r, ») tables are completed, save a great 
deal of calculation in the finding of any series of y-functions ; the inter- 
polated values must then be substituted in equation (iv.) to find x (7, 9), 
and this value substituted in (iii.) will give F (rv, 7). It will be found that 
this needs only a moderate amount of arithmetic, but if it nas to be done 
for a considerable number of frequency curves, the statistician may still 
reasonably demand the completion of the F (7, 7) tables themselves. 
APPENDIX. 
Tables of x-functions (x1) X39 X51 and x7): 
These tables have been calculated by Miss A. Lee, Mr. G. U. Yule, 
Dr. C. E. Cullis, and Mr. Karl Pearson, and the independent values 
thus obtained used for the verification and correction of the tables 
originally provided by Mr. H. F. Harris. 
The figures in brackets will generally only be required to determine 
the accurate seventh figure in the value of the y-function or its differences. 
The differences in the higher values of y have been found by calculating 
x to eleven figures and then dropping the last two. On this account it 
will be found that the tabulated differences do not in the bracketed figures 
always agree in the last place with the results obtained by subtracting 
the tabulated y’s. 
Two differences will always suffice to calculate y3, x;, and y, to seven 
places, and even with x, two differences will very rarely give a unit error 
in the last place of figures, while the use of the third difference would 
amply suftice for all seven places. 
TABLE OF VALUES OF y. 
% log (4X1) X1 Ay As 
o 
0 2°7958800 —-0625000(00) = -- 
1 2°7957037 — -0624746(30) +  253(70) = 
2 2°7951742 —-0623985(00) + 761(30) + 507(60) 
3 2-7942911 — ‘0622717(57) + 1267(43) + 606(13) 
4 27930532 —-0620945(14) + 1772(48) +505(00) 
5 2-7914590 —+0618670(00) + 2275(14) + 502(71) 
6 27895066 —-0615894(84) + 2775(16) + 500(02) 
7 27871935 —:0612623(30) + 3271(54) + 496(38) 
8 27845168 —-0608859(00) + 38764(30) +492(76) - 
9 27814731 —+0604607(00) + 4252(00) + 487(70) 
10 27780586 —+0599872(00) + °4735(00) + 483(00) 
11 27742688 —-0594660(14) + 5211(86) +476(86) 
12 27700986 —-0588977(27) + 5682(87) +471(01) 
13 2°7655426 — -0582831(00) + 6146(27) + 463(40) 
14 27605945 —0576228(13) + 6602(87) + 456(60) 
15 27552476 —-056917 7(40) + 7050(73) +447(86) 
16 27494942 —-0561686(64) + 7490(76) + 440(03) 
17 2°7433260 — -0553765(50) + 7921(14) + 430(38) 
18 27367341 —-0545423(75) + 8341(75) + 420(61) 
19 2-7297083 —+0536671(25) + 8752(50) + 410(75) 
20 27222378 —*0527518(61) + 9152(64) + 400(14) 
