Mr. W. Dittmar on Graphical Interpolation. 139 
out the intervention of logarithms. It will therefore be especi- 
ally useful in chemical calculations, as, for instance, in converting 
specific grayities into specific volumes, for reducing per-centage 
compositions to the unit of weight of one constituent, &c. The 
general applicability of this method is evident. For the sake of 
illustration I append figs. 2 and 8, giving respectively the be- 
ginning of a general interpolation- and of a densimetric Table. 
The mode of construction and use of these Tables will be under- 
stood from the description given of the Table of logarithms. 
XXII. On Graphical Interpolation. By W. Dittmar, Assistant 
in the Laboratory of Owens College, Manchester*. 
et principles laid down in the preceding article for the 
construction of numerical tables may also be employed for 
the purpose of carrying out graphical interpolations. Let us 
suppose that the corresponding values x¥o, 2)Y1 %eYo-++ be- 
longing to an unknown function y=/(z) are given by obser- 
vation, and that it is required to complete the series of vari- 
ables. It is clear that the direct results of observation may be 
registered in a graphical table in the manner described above. 
For this purpose itis only necessary to draw a straight line, and 
to construct on one side of it a scale with a constant unit of 
length, the points of which are considered as representatives of 
the values of y, while on the other side of the line marks made 
opposite to the points 7, ¥;, Ya,--. are taken as symbols of the 
respective values of 2, 1. €. %, 2, Zq,...&c. The question now 
is, how can the gaps on the w-scale be filled up by graphic inter- 
polation? This may be accomplished in the following way :— 
When there are reasons for supposing that f(z) does not differ 
much from a linear function, all the divisions on the x-scale may 
be made equal to one another, and each so long that the points 
corresponding to 2%, #,, &c. coincide as nearly as possible with 
those signifying respectively 7, y,, &c. on the y-scale. This is 
best done by dividing the distance between the two furthest 
points on the z-scale into the requisite number of equal parts, 
drawing lines from the points thus obtained to one point situated 
at some distance, and by moving the y-scale along in this system 
of lines parallel to the line dividedy, till a position is found in 
which the points of intersection of the radu with the y-scale 
yield an w-scale which agrees as closely as possible with the 
observed values. 
* Communicated by the Author. 
7 A sharp-edged drawing measure is most conveniently employed for 
this purpose. 
