328 ON THE MOTrON OF 



pointed out without being able to demonstrate, are rigorously 

 deduced from the linear differential equations in which they 

 are comprised ; and the beautiful theory of these equations, 

 including their complete integration in a finite series of the 

 multiples of sines of arcs proportional to the time, is, develo- 

 ped and explained with admirable skill. An easy application 

 of the principles of this theory solves the problem of the os- 

 cillation of a heterogeneous circle within a circle, without 

 friction, or what is essentially the same question, of any solid 

 upon any suitable surface, the plane of motion being invaria- 

 ble ; as for instance a spherical segment in a spherical cup, 

 supposing no whirling to take place, or a pendulum with 

 cylindrical pivots working in cylindrical collars, which is the 

 form in which the problem is proposed by Euler himself. 

 When the friction prevents all sliding, the oscillation is sin- 

 gle, and is determined without reference to the theory just 

 mentioned. The effect which this friction has in diminish- 

 ing the time of a pendulum's vibrations, (along with a va- 

 riety of other circumstances necessary to take into the ac- 

 count when the appareil of Borda is employed) has been also 

 calculated by Laplace in a paper on the seconds' pendulum 

 inserted in the Connaissance des Tews for 1820. His me- 

 moir is remarkable for the subtlety of the analysis, rendered 

 necessary by the multitude of the considerations included in 

 his calculus, but when he mentions the effect of friction 

 without sliding as a singular and interesting result to which 

 he had arrived, he is evidently not aware of the formulas of 

 Euler and John Bernoulli, from either of which the same in- 

 ference may readily be drawn. 



In the Ada Petropolitana for 1782, one year before his 

 death, Euler resumes the investigation of the problem he 

 had considered in -his Theoria motus corporum rigidorum. 

 This problem, which consisted, as I have already mentioned, 

 in determining the motion of a heterogeneous sphere along a 

 horizontal plane, is called by Euler himself, qusestio maxima 

 ardua, and is regarded by him as inaccessible by the methods 

 then in use, except in the case in which the centres of gravity 



