1876.] Differential Equations of any Order. 275 



that another (a ^/-auxiliary) shall have displacement always equal to 



(y'-y)f{(x'-*f+(y'-y) 2 }, 



that another (an ^-auxiliary) shall have displacement equal to 



{x "-ce)f{(x"- X ? + (y"- y y), 

 and so on. 



Then connect the first globe-frame o£ the first double integrator, so 

 that its displacement shall be equal to the sum of the displacements of 

 the ^-auxiliaries ; that is to say, to 



(x'-x)f{(x'-xy + (y'-yf} 

 ' + (x'-x)f{x"-xY + (y"-yf} 

 + &c. 



This may be done by a cord passing over pulleys attached to the 

 x- auxiliaries, with one end of it fixed and the other attached to the globe- 

 frame (as in my tide-predicting machine, or in Wheatstone's alphabetic 

 telegraph-sending instrument). 



Then, to begin with, adjust the second globe-frames and the second 

 cylinders to have their displacements equal to the initial velocity-compo- 

 nents and initial coordinates of % particles free to move in one plane. 

 Turn the machine, and the positions of the particles at time t are shown 

 by the second cylinders of the several double integrators, supposing them 

 to be free particles attracting or repelling another with forces varying 

 according to any function of the distance. 



The same may clearly be done for particles moving in three dimensions 

 of space, since the components of force on each may be mechanically 

 constructed by aid of a cam-surface whose equation is 



and taking rj for the distance between any two particles, and 



%=x' —x 

 or =y'-y 

 or =x" — x, &c. 



Thus we have a complete mechanical integration of the problem of finding 

 the free motious of any number of mutually influencing particles, not 

 restricted by any of the approximate suppositions which the analytical 

 treatment of the lunar and planetary theories requires. 



