Miscellanea 
101 
Let us first find a value for .*; = 45, _y = 17, ;=12 ; this only involves two variables, and fig. 7 
tells us to interpolate between 45, 10, 5 and 45, 15, 10, etc. ; hence the interpolated value required 
is obtained by ordinary interpolation in the following way : 
A 
A-' 
7-24 
13-80 
12-25 
13-61 
21-04 
26-05 
26-86 
47-09 
52-91 
100-00 
therefore value is 
7-24 + 1-4X 13-80 + ^^ 12-25 -i^-^^i't) ^'^^ 13-61=29-13, 
while with only three terms we should have obtained 29-89. If we had used the ordinary double- 
interpolation method with the terms 21-04, 28-20, 47-09 and 35-03, we should have obtained 
30-28, and as the true result is 29-48, we have in this case obtained a better approximation by 
the "interpolation-line" method. The data are not good for intei-polation because the difterences 
do not decrease, but this is no disadvantage for our present purpose. 
As a second example we will take x = QO, j/=18, 2=11, and we can use 14-30, 13-33 and 9-71 
and obtain 
14-30 - -8 X -97 + ^-^^ X 2-65 = 1 3-73, 
and the true value is 13-69. Two terms only would have given 13-58. With a little practice the 
values to be used can be easily picked out. 
We will take as our final examples two three-variable cases. To obtain ,v = 49, .y = 18, s= 14 we 
can use 45, 20, 10 and 55, 15, 20 and take two-fifths of the difference, i.e. 28-20 -f* (31-39 - 28-20) 
= 29-48, or we can use 50, 20, 15 and 45, 10, 10 thus getting 31-21 while if we add 55, 30, 20 which 
is 70-02 we get 29-93 ; the true value is 30-31. To obtain x=b% _y = 14, s= 14 we can use 50, 10, 15 
and 60, 30, 10 and obtain 21-10, or we can take 50, 15, 15 and 60, 10, 10 but the former is 
distinctly preferable and if we add the term 70, 50, 5 which is 17-63 it becomes pi-actically 
exact. 
The latter arrangement, however, shows the method at its worst. The differences run 
extremely awkwardly and the interpolation by firet differences leads to an luitrustworthy result. 
When using the " interpolation-line " method we are, as it were, picking out a level piece of 
ground on our surface before interpolating ; if we choose an uneven piece of ground, i.e. if the 
differences are large, the result will be inaccurate. 
5. Connection of Method with the Construction, of Tables. An interesting point of some 
practical importance may now be examined. If we were to make a new two-variable table, what 
would be the best interval to use if we intended to apply the "intei-polation-line" method 1 
We must clearly use an interval so that the terms used in the interpolations are not far 
apart, and in order to answer the question it becomes necessary to consider what intervals are 
most satisfactory from this \)omt of view. Wo will first see how the various interpolations can 
be built up and will take the interval 5 to start with. The terms we require to find by interpola- 
tion are 1, 1 ; 1, 2 and 2, 2, because 1, 3 is just the same as 1, 2 but with a different starting 
point, for it is merely distance 1, 2 from the given term 0, 5. There are therefore only two lines 
necessary, one symbolised by 1, 1 and showing interpolation Ijetween terms like 0, 0 and 5, 5, 
and the other by 1, 2. The former gives 2, 2 ; wo should have to take two-fifths of the distance 
between the 0, 0 and 5, 5 given terms instead of the one-fifth that gave 1, 1. Now let us take 7 
as an interval. The terms wanted are 1, 1 ; 1, 2 ; 1, 3; 2, 2; 2, 3; 3, 3. The 1, 1 line gives 1, 1 ; 
2, 2 and 3, 3 ; and the 1, 2 line gives 1, 2 and 2, 4, but the second of these is the same as 2, 3 with 
a different starting-point, and we only therefore require two lines. The idea is easily continued. 
