VI. Mechanical integration of the general linear differential equation of any order with variable coefficients

Take any number i of my brother’s disk-, globe-, and cylinder-integrators, and make an integrating chain of them thus:—Connect the cylinder of the first so as to give a motion equal to its own to the fork of the second. Similarly connect the cylinder of the second with the fork of the third, and so on. Let g1, g2, g3, up to gi be the positions of the globes at any time. Let infinitesimal motions P1dx P2dx P3dx .... be given simultaneously to all the disks (dx denoting an infinitesimal motion of some part of the mechanism whose displacement it is convenient to take as independent variable). The motions (dk1, dk2, . . . dki) of the cylinders thus produced are dk1=g1 P1dx, dk2=g2 P2dx,... dki=gi Pidx... (1) But, by the connexions between the cylinders and forks which move the globes, dK1 = dg2, dK2 = dg3, . . . dKi-1 = dgi , and therefore