270 On Differential Equations of the Second Order. [Feb. 3, 



will give out continuously the value of u 2 . Use again the same process 

 with u 2 instead of u v and then m 3 , and so on. 



After thus altering, as it were, n x into u 2 by passing it through the 

 machine, then u 2 into u 3 by a second passage through the machine, and so 

 on, the thing will, as it were, become refined into a solution which will 

 be more and more nearly rigorously correct the of tener we pass it through 

 the machine. If u i+1 does not sensibly differ from u it then each is sen- 

 sibly a solution. 



So far I had gone and was satisfied, feeling I had done what I wished 

 to do for many years. But then came a pleasing surprise. Compel 

 agreement between the function fed into the double machine and that 

 given out by it. This is to be done by establishing a connexion which 

 shall cause the motion of the centre of the globe of the first integrator of 

 the double machine to be the same as that of the surface of the second 

 integrator's cylinder. The motion of each will thus be necessarily a 

 solution of (1). Thus I was led to a conclusion which was quite unex- 

 pected ; and it seems to me very remarkable that the general differential 

 equation of the second order with variable coefficients may be rigorously, 

 continuously, and in a single process solved by a machine. 



Take up the whole matter ah initio : here it is. Take two of my 

 brother's disk-, globe-, and cylinder-integrators, and connect the fork 

 which guides the motion of the globe of each of the integrators, by 

 proper mechanical means, with the circumference of the other inte- 

 grator's cylinder. Then move one integrator's disk through an angle =x, 

 and simultaneously move the other integrator's disk through an angle 

 always =\*~£doc, a given function of x. The circumference of the 

 second integrator's cylinder and the centre of the first integrator's globe 

 move each of them through a space which satisfies the differential 

 equation (1). 



To prove this, let at any time g v g 2 be the displacements of the centres 

 of the two globes from the axial lines of the disks ; and let dx, Vdx be 

 infinitesimal angles turned through by the two disks. The infinitesimal 

 motions produced in the circumferences of two cylinders will be 



g x dx and gj?dx. 



But the connexions pull the second and first globes through spaces 

 respectively equal to those moved through by the circumferences of the 

 first and second cylinders. Hence 



g iC lx=dg 2 , and g 2 Vdx=dg x ; 

 and eliminating g 2 , 



dx\pdx) Jv 



which shows that g x put for u satisfies the differential equation (1). 



