1876.] Differential Equations of any Order. 273 



Until it is desired actually to construct a machine for thus integrating 

 differential equations of the third or any higher order, it is not necessary 

 to go into details as to plans for the mechanical fulfilment of condition (7) ; 

 it is enough to know that it can be fulfilled by pure mechanism working 

 continuously in connexion with the rotating disks of the train of inte- 

 grators. 



Addendum. Eeceived February 10_, 1876. 



The integrator may be applied to integrate any differential equation of 

 any order. Let there be i simple integrators ; let &\, g v k x be the dis- 

 placements of disk, globe, and cylinder of the first, and so for the others. 

 "We have 



ch die 



Now by proper mechanism establish such relations between 



x v 9v K v x v &> &c - 

 that 



f m (*v9v K i> x » • • •)=°» 



/ (2) (>i>#i>Ki>^ •••)=(>, 



(2i — 1 relations). 



This will leave just one degree of freedom ; and thus we have 2i—l 

 simultaneous equations solved. As one particular case of relations take 



Xf=xj=z . . . (i— 2 relations), 

 and 



g.y=K v # 3 =K 2 ' & c< (* — 2 relations); 

 so that 



_cl l tc dh^fi 



Thus one relation is still available. Let it be 



f(v,9v92>'-f/i> »0 = - 

 Thus the machine solves the differential equation 



-/ d*u d l ~ l u du \ A , ... s 



f ( '• 12 SF? • ■ ' W u ) - ° (puttlDg u for ^ 



Or again, take 2i double integrators. Let the disks of all be connected 



