1876.] On Differential Equations of the Second Order. 269 



V. " Mechanical Integration of the Linear Differential Equations 

 of the Second Order with Variable Coefficients." By Prof. 

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

 28, 1876. 



Every linear differential equation of the second order may, as is 

 known, be reduced to the form 



d /l du\ /1 x 



MpsJ =w (1) 



where P is any given function of x. 



On account of the great importance of this equation in mathematical 

 physics (vibrations of a non-uniform stretched cord, of a hanging chain, 

 of water in a canal of non-uniform breadth and depth, of air in a pipe of 

 non-uniform sectional area, conduction of heat along a bar of non-uniform 

 section or non-uniform conductivity, Laplace's differential equation of 

 the tides, &c. &c), I have long endeavoured to obtain a means of facili- 

 tating its practical solution. 



Methods of calculation such as those used by Laplace himself are 

 exceedingly valuable, but are very laborious, too laborious unless a serious 

 object is to be attained by calculating out results with minute accuracy. 

 A ready means of obtaining approximate results which shall show the 

 general character of the solutions, such as those so well worked out by 

 Sturm*, has always seemed to me a desideratum. Therefore I have made 

 many attempts to plan a mechanical integrator which should give solu- 

 tions by successive approximations. This is clearly done now, when we 

 have the instrument for calculating \f{po) \p(x) dec, founded on my brother's 

 disk-, globe-, and cylinder-integrator, and described in a previous com- 

 munication to the Royal Society ; for it is easily proved t that if 



' J ° J ° 1 (2) 



W 3=j;p(c-j;v^)^J 



&c, 



where u x is any function of ac, to begin with, as for example u x =x ; then 

 u 2 , u 3 , &c. are successive approximations converging to that one of the 

 solutions of (1) which vanishes when a?=Q. 



Now let my brother's integrator be applied to find C— • f* u. L dx, and let its 

 result feed, as it were, continuously a second machine, which shall find 

 the integral of the product of its result into P dec. The second machine 



* " Memoire sur les equations differentielles lineaires du'; second ordre," Liouville's 

 Journal, vol. i. 1836. 



t Cambridge Senate-House Examination, Thursday afternoon, January 22nd, 1874. 



