1876.] On Differential Equations of any Order. 271 



The machine gives the complete integral of the equation with its two 

 arbitrary constants. For, for any particular value of x, give arbitrary 

 values Gr x , Gr 2 . [That is to say mechanically ; disconnect the forks from the 

 cylinders, shift the forks till the globes' centres are at distances Gr l5 Gr 2 

 from the axial lines, then connect, and move the machine.] 



"We have for this value of x, 



that is, we secure arbitrary values for g x and -p by the arbitrariness of 

 the two initial positions Gt v Gr 2 of the globes. 



VI. " Mechanical Integration of the general Linear Differential 

 Equation of any Order with Variable Coefficients.'''' By Prof. 

 Sir William Thomson, LL.D., F.R.S. Received January 28, 

 1876. 



Take any number * of my brother's disk-, globe-, and cylinder-inte- 

 grators, 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 g v g 2 , g 3i up to g i} be the positions f of 

 the globes at any time. Let infinitesimal motions P/fo?, ~P 2 dx, ~P 3 dx, .... 

 be given simultaneously to all the disks (dec denoting an infinitesimal 

 motion of some part of the mechanism whose displacement it is convenient 

 to take as independent variable). The motions (dK v dic 2 , . . . d^) of the 

 cylinders thus produced are 



dK 1 ^=g 1 'P l dx 3 dic 2 =g 2 ~P 2 dx, . . ,dK i =g i ^ i dx\ . . . (1) 



But, by the connexions between the cylinders and forks which move the 

 globes, dK^=dg 2i dic 2 =dg 3 , . . . dic i _- [ = dg i i and therefore 



and I . . (2) 



dK=g r V x dx, dK 2 =g 2 T? 2 dx, . . . e^=# V t dx. J 

 Hence 



_ 1 d 1 d 1 d 1 dx { ,„> 



fJl ~F x TxT 2 dx''''YZidx¥idx~ w 



Suppose, now, for the moment that we couple the last cylinder with the 



* For brevity, the motion of the circumference of the cylinder is called the cylinder's 

 motion. 



t For brevity, the term "position" of any one of the globes is used to denote its 

 distance, positive or negative, from the axial line of the rotating disk on which it 



