G. K. Gilbert—Special Processes of Research. 467 
The next higher division of the graphic method is homol- 
ogous with the algebraic equation containiny three unknown 
quantities, and is employed in the discussion of a phenomenon 
with reference to its simultaneous relations to two other phe- 
nomena. If in an equation between three variables we make 
one of the variables successively equal to 0, 1, 2, 3, ete, we 
obtain a series of equations between the remaining two varia- 
bles. Hach of these equations is the equivalent of a curve, 
and we may plat the equivalent curves upon the same sheet of 
paper, referring them to the same codrdinate axes. They con- 
stitute a system, and they collectively express, in connection 
with the other systems of lines on the paper, the original equa- 
tion between three variables. If the sheet upon which they 
are platted is ordinary engineer’s section paper, then we may 
Say that the vertical lines of the paper represent equidistant 
values of one variable, the horizontal lines equidistant values 
of a second, and the system of curves equidistant values of the 
third. Definite values being assigned to either two of the 
variables, the corresponding value of the third may be found 
by inspection. This principle is extensively applied in abridg- 
- ing the labor of computation where a large number of results 
are to be derived by means of the same formula. 
It is possible that the nature of this system of curves will be 
more clearly apprehended if we approach it by a route involv- 
ing more of geometry and less of algebra. An equation between 
three variables is the algebraic equivalent of a surface, either 
plane or curved. Let us think of such a surfaceas forming the 
uneven top of a solid body whose base is flat and square, and 
whose sides are vertical. Horizontal distances along the sides 
in one direction correspond to values of one variable, and in 
the transverse direction to those of another, while vertical dis- 
tances correspond to values of the third. Now intersect the 
solid by a system of horizontal planes, each one at a height 
above the base determined by an integral value of the third 
variable. Hach plane will cut the upper surface in a different 
curved line, and each of these lines will be a contour of the 
surface. Now move all these lines vertically downward to the 
basal plane, and there is formed on that plane a contour map of 
the upper surface of the solid. This contour map is identical 
with the system of curves obtained through the algebraic ma- 
nipulation of the equation. The relation between the three 
variables has thus three separate but equivalent expressions 
(from a‘certain system of stations) in a single year. In fig. 4, curve A, each ordi- 
nate represents the mean of the numbers observed in three consecutive years. 
That is to say, Professor Loomis ‘‘smoothed” the curve by a process making 
each ordinate an equal function of the observations of the corresponding year, 
the preceding year and the following year, while I employed the unadjusted data 
of observation. 
