140 Mr. W. Dittmar on Graphical Interpolation. 
If a satisfactory result cannot be obtained im this way, it is 
best to try whether /(z) can be practically represented by an 
expression of the form A+ Bz+Ca?, where A, B, and C signify 
constants. If this be the case, we have* 
Y=AS BOTs ee ies ss. 8 oc 
y— Ay=A-+ B(@—Az) + C(e#—Az)?. 
Ay=BAz-—C(Az)?+(2CAz)z.. . . . » (IL) 
Ax Yen —BAx—C(Aa)?+ (2CAz) G + = (IIL) 
2,—X, 2 
Comparing equation (III.) with (II.), we see that In In Nz 
Lx, 
is equal to this Ay, the graphical representative of which is con- 
tained between the two points in the 2-scale corresponding to 
x +x, + Az Bidets tenn At _ A, 
—— ease : 
the numbers 
From the equations (II.) and (III.) the following method for 
constructing the z-scale may be found :—Combine the observed 
pairs of variables by twos, and find from every combination, with 
the help of equation (III.), a certain Ay, the graphical represen- 
tative of which is contained between the two points which in the 
z-scale mean z—~Az and x ‘Then construct a rectangular 
system of coordinates, and represent the values of x thus obtained 
(with an arbitrary unit of length) as abscissc, the corresponding 
values of Ay as*ordinates, using for the latter that length as unit 
which represents Ay=1 in the y-scale. Next draw a straight 
line which passes as nearly as possible through the extreme 
points of the ordinates. The ordinates of this line correspond- 
ing respectively to Aw, 2Az, 3Az,..., when put together in the 
right order, give the required z-scale. In order to obtain exact 
results, it is advisable to choose first such a large value for Ax 
that only a few points of the w-scale are obtained, and to deter- 
mine the intermediate pomts by new constructions. As soon as 
so many points are determined that two successive intervals do not 
differ perceptibly in length, the subdivisions of each interval may 
be made equal to one another. Should the indications of a scale 
thus obtained not agree quite satisfactorily with the observed 
data, it may often be improved by slightly changing the unit of 
length used in the construction, and by altering its position with 
respect to the y-scale. A convenient method for doing this has 
been already described. 
* g and y mean any values of the variables belonging together; xp, yp, 
and @n, Yn mean particular pairs of variables; Av stands for the constant 
numerical difference corresponding to one division in the a-scale; Ay for 
the corresponding variable increment of y. 
