e displacements y of
the points C (fig. 24) each spring is attached to a lever EC, fulcrum E. To
this again a long rod FG is fixed by aid of a joint at F. The lower end of
this rod rests on another lever GP, fulcrum N, at a changeable distance y"
= NG from N. The elongation y of any spring s can thus be produced by a
motion of P. If P be raised through a distance y', then the displacement y
of C will be proportional to y'y"; it is, say, equal to [mu]y'y" where [mu]
is the same for all springs. Now let the points C, and with it the springs
s, the levers, &c., be numbered C_0, C_1, C_2 ... There will be a
zero-position for the points P all in a straight horizontal line. When in
this position the points C will also be in a line, and this we take as axis
of x. On it the points C_0, C_1, C_2 ... follow at equal distances, say
each equal to h. The point C_k lies at the distance kh which gives the x of
this point. Suppose now that the rods FG are all set at unit distance NG
from N, and that the points P be raised so as to form points in a
continuous curve y' = [phi](x), then the points C will lie in a curve y =
[mu][phi](x). The area of this curve is

[mu] [Integral,0:c][phi](x)dx.

Approximately this equals [Sigma]hy = h[Sigma]y. Hence we have

[Integral,0:c][phi](x)dx = h/[mu] [Sigma]y = ([lambda]h/[mu])z,

where z is the displacement of the point B which can be measured. The curve
y' = [phi](x) may be supposed cut out as a templet. By putting this under
the points P the area of the curve is thus determined--the instrument is a
simple integrator.

The integral can be made more general by varying the distances NG = y".
These can be set to form another curve y" = f(x). We have now y = [mu]y'y"
= [mu] f(x) [phi](x), and get as before

[Integral,0:c]f(x) [phi](x)dx = ([lambda]h/[mu])z.

These integrals are obtained by the addition of ordinates, and therefore by
an approximate method. But the ordinates are numerous, there being 79 of
them, and the results are in consequence very accurate. The displacement z
of B is small, but it can be magnified by taking the reading of a point T'
on the lever AB. The actual reading is done at point T connected with T' by
a long vertical rod. At T either a scale can be placed or a drawing-board,
on which a pen at T marks the displacement.

If the points G are set so that the distances NG on the different levers
are proportional to the terms of a numerical series

u_0 + u_1 + u_2