z = [Sigma]y / (n a/b + b/a K/k) = [Sigma]y / (n (a/b + l/L)).
This shows that the displacement z of B is proportional to the sum of the
displacements y of the tops of the small springs. The arrangement can
therefore be used for the addition of a number of displacements. The
instrument made has eighty small springs, and the authors state that from
the experience gained there is no impossibility of increasing their number
even to a thousand. The displacement z, which necessarily must be small,
can be enlarged by aid of a lever OT'. To regulate the 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.