96 
Miscellanea 
2. The Interpolation Line Method. We may now attack the theory of this second method 
in greater detail. 
Let us assume that from the following tabulated terms it is required to find 
"0,(l,0, . ; W,., 5, , W2)-, 2s, 2(, ... ) CtC, 
■"O, .<!,(, ...5 W,.^ 2s, 2(, ... ) ''2)',3.'i,3(, ..1 etc., 
etc., etc. 
This scheme is general and there is no loss of generality in assuming that the interpolated 
values are for integers, i.e. if the function is tabulated for x=0, 5, 10, etc., y=0, 8, 16, etc., 
2=0, 2, 4, etc., the interpolated values required would be for x=^\, 2, 3, 4, etc., y=\, 2, 3, 
2 = 1, 3, 5, ... ; because if fractional values are required we can assume that the ^s, y'a, z's, etc. 
proceed by larger differences. 
h ^ h 
Now in order to get «,! we take - and use the formula (H-A)''Mo=%, or we can take — and 
Ji_ k 
use (1 +A')2''?<o=?fft, where A=m,.-?<o f^nd A'=?<o,. -w-o- Again to get Mj,^^ we take (1 + A)«Mo oi" 
(l + A')2«?<o 01" etc., where A and A' have similar meanings. 
But if ^' be used for a two-variable, then the interpolated function found could be 
?f , /i , and therefore in order to get terms from which interpolation can be made we 
A 
must use, not Ui-g, etc., but «iki, u,.^^sfi, so that when we take (l + Aa)''"Ma:y we shall get 
It i.s fairly clear that if we can fix either ra or s/3 the other can be obtained easily, and a little 
consideration of the way the ra and s/3 arise will show the reader that the term can be fixed by 
the least common multiple of k, r, k and s. 
A numerical example will make this clearer. Let «n,05 ^5,8) «in,ioi etc. be given and assume 
that we require ?;2,.j ; then the l.c.m. of 2, 3, 5, 8 being 120 we must interpolate between 
%0) %o,i2o> Mi(io,240) for having fixed the 120, the proportion to be taken is ]fo = 4\) and 
2x40=80. If a third variable is introduced the same method can be used, and if in our last 
numerical example we take Mo,o,0) %,8,i0) ^10,16, 20 > ••• given, and require M2,3,7) we can use 
'<o,o,0) ^240,360,840) ^480, 720, 1080 1 etc. The numerical work would be as follows in an imaginary 
example : 
A A2 
Mo, 0,0 = 4872 -2760 +1656, 
"240,300,840=2112 - 1104, 
■"480,720,1680= 1008. 
.-. M2,3,7 = 4872- jig. 2760- l-yij. fig. 1656 
= 4832. 
There are, of course, many objections to such a method in practice when it entails differences 
in the independent variable, such as 240, 360 and 840, but it can often be simplified considerably ; 
for instance, in the last example, if we imagine the given values to extend to negative values of 
X, y and 2, we can start from 'ifj^cio and seek M2,3,7- We notice that our fractions are now 
5)8)^ and the l.c.m. is reduced to 120. The terms to be used are Mj.o.ioi "-116,120, -110 and 
W-235,24o,-230j and the value of ?«2,3,7 is 
W5, o, 10 + (t<- nr,, 120, - 110 - "5, o, 10) - i • 5^ • Iff (second diff". ). 
In this way we can sometimes diminish the range of values considerably. Even here however a 
large table would be required for practical work, but it must be borne in mind that a three- 
variable table is very seldom made, and a two-variable table is frequently simplified by both the 
.r's and ?/'s ^jroceeding by the same difi'erence. The difficulty is however always present to 
some extent and it would be impossible to find values of u corresponding to a number of decimal 
places in x^y, ... , though the method might even in these cases be of help as a rough check. 
