100 
Miscellanea 
In this case in order to obtain an interpolated value for ««2,i we require ?to,0) "6,4 ^^d ?t_6, -4 ; 
cleai'ly a less useful tabulation than differences of 3 in both x and y, and not much improvement 
(sometimes it is worse) than differences of 4 in both x and y. The lesson to be learnt from this 
is that if one can choose one's differences in tabulation it is best to take the same difference in 
both X and y. 
Difference in x = 10. 
., » y =10- 
The diagram in this case becomes rather troublesome, but the following table answers the 
same purpose. 
X 
y 
Interpolate betw 
Ben 
11 
Taking — of distance 
where n is 
1 
1 
0, 
0 
and 
10, 
10 
I 
1 
1 
2 
0, 
0 
and 
10, 
20 
II 
1 
1 
3 
0, 
0 
and 
10, 
30 
III 
1 
1 
4 
10, 
10 
and 
-20, - 
10 
IV 
3 
1 
5 
0, 
0 
and 
10, 
50 
V 
1 
2 
2 
0, 
0 
and 
10, 
10 
I 
2 
2 
3 
0, 
0 
and 
20, 
30 
IV 
1 
2 
4 
0, 
0 
and 
10, 
20 
II 
2 
2 
5 
0, 
0 
and 
20, 
50 
VI 
1 
3 
3 
0, 
0 
and 
10, 
10 
I 
3 
3 
4 
0, 
10 
and 
10, - 
10 
II 
3 
3 
5 
0, 
-10 
and 
10, 
40 
V 
3 
4 
4 
0, 
0 
and 
10, 
10 
I 
4 
4 
5 
10, 
-10 
and 
-10, 
40 
VI 
3 
5 
5 
0, 
0 
and 
10, 
10 
I 
5 
This would of course be applicable where one decimal place is required and the tabulation is for 
integral values, but as will be noticed immediately, the number of independent " interpolation- 
lines " (see Roman numbers) is very great, and it is not very difficult to show that the decimal 
system is by no means the best method of tabulating functions. We shall however return to 
this after showing the application of the above diagrams to numerical work. 
4. Numerical Examples. The following table gives the values of a function of three- 
variables, which will enable us to give a few examples. 
x = 
= 45 
x = 
= 50 
x = 
= 55 
.T = 60 
y 
y 
y 
y 
2 
10 
15 
20 
25 
10 
15 
20 
25 
10 
15 
20 
25 
10 
15 
20 
25 
30 
5 
10 
15 
20 
7-24 
16-06 
26-76 
39-73 
9-51 
21-04 
.35-03 
52-12 
12-74 
28-20 
47-09 
70-49 
17-71 
39-36 
66-18 
100-00 
5-74 
12-70 
21-08 
31-13 
7-39 
16-27 
26-94 
39-81 
9-60 
21-11 
35-01 
51-99 
12-74 
28-11 
46-90 
70-27 
4-69 
10-34 
17-09 
25-09 
5-93 
13-01 
21-41 
31-39 
7-53 
16-45 
27-06 
39-84 
9-65 
21-13 
34-94 
51-84 
3-95 
8-69 
14-30 
20-86 
4-94 
10-77 
17-61 
25-61 
6-14 
13-33 
21-74 
31-70 
7-66 
16-62 
27-23 
39-95 
9-71 
21-19 
34-97 
51-74 
