[ 31 ] 

 III. Note on Foucault's Gyroscope. By M. 3. Bertrand*. 

 HE ingenious apparatus to which M. Leon Foucault has 



T 



given the name of gyroscope, is now well known to all 

 philosophers. There is no necessity, therefore, for me here to 

 redescribe the same ; it will be sufficient to recall to mind that, 

 essentially, it consists of a solid of revolution turning rapidly 

 around its axis of figure at the same time that the latter is 

 obliged, by the nature of the system, to remain in a plane fixed 

 with respect to the earth. The movement of the axis of the 

 gyroscope in this plane, however, is perfectly free. 



The explanation, almost intuitive, of the observed phsenomena 

 is to be sought in the principles discovered by Poinsot, and the 

 following note is but a corollary to the admirable memoir written 

 by this celebrated geometer twenty years ago, and published 

 entire in the 16th volume of the first series of Liouville's Journal. 



It is well known that Poinsot regards each molecule of a 

 moving body as animated by a force equal to the product of its 

 mass into its velocity. All the forces which at a given moment 

 animate the molecules of a moving solid body, may be composed 

 by the rules of statics, and reduced to a force and a couple ; if 

 the solid body is free, and not solicited by any exterior force, 

 this resultant force and couple are invariable. But if the influ- 

 ence of exterior forces is superadded to that of inertia, the 

 system of forces which animate the Jiody at the expiration of an 

 infinitely small interval of time dt, may be considered as com- 

 posed of two others ; first, the system of finite forces which 

 animated the body at the commencement of the interval under 

 consideration; and secondly, the system of exterior forces which 

 have acted on the body, each multiplied by the magnitude dt of 

 that interval. * 



This fundamental principle conducted Poinsot to his most 

 elegant theorems, and as will be seen, it suffices for the complete 

 explanation of the phjcnomena discovered by Foucault. 



We shall suppose the apparatus to be so disposed that the 

 axis of rotation, which is the axis of symmetry of the rotating 

 body, is compelled to remain in a ])hnie P, fixed in relation to 

 the earth. Let o be the centre of the instrument, conceived as 

 fixed, and let us examine solely the motion of the system 

 around this point, reducing, consequently, all the forces to the 

 couples which they produce. 



Let cjA be the actual ])osition of the axis of rotation in the 

 plane 1*, and o\. the parallel to the earth's axis drawn through 

 the point o. 



* I'Vdih tlic Journnl ilc ]\I(ithemaliques puns ei appUqucs, 2iul scries, 

 vul. i. p. '.iJU. 



