32 M. J. Bertrand on Foucaiilt's Gyroscope. 



In order that the axis oK may remain apparently at rest in 

 the plane P, it must in reality turn round o\ with a velocity 

 equal to that of the earth, and in twenty-four hours describe a 

 cone of revolution. Let oA! be a position upon this cone infi- 

 nitely close to oA. The couple which animates the gyroscope 

 turning, during the first instant, around oA, has its axis directed 

 along this line, and equal to the product of the moment of iner- 

 tia fjb into the angular velocity m. In order that this axis, which 

 we will represent by oG, may, during the following instant, 

 become oG' (directed along oA'), the system, during the infi- 

 nitely small period dt, must have been solicited by a couple 

 directed along GG', and having an intensity represented by 



GG' 



dt ■ 



Now the only action directly experienced by the instrument is 

 the reaction of the fixed plane P ; this reaction can only produce 

 forces perpendicular to the plane, and, consequently, a couple 

 with its axis situated in this plane. The line GG', therefore, 

 must be parallel to the plane P, and for this reason P must be a 

 tangent plane to the cone, and hence perpendicular to the plane. 

 We have then this first theorem : — 



The axis of the gyroscope being compelled to remain in a plane 

 P, it cannot remain in equilibrium unless it coincides tvith the 

 projection uf the parallel to the earth's axis upon the fixed plane. 



When tliis coincidence does not exist at the commencement, 

 relative equilibrium is impossible, and the instrument is set in 

 oscillation, the laws of which we must calculate. 



We may at once remark, that, whatever may be the initial 

 position oA of the axis, it is allowable to apply to the instru- 

 ment two equal and contrary couples, one of wbich would, alone, 

 retain the axis in a state of apparent repose without changing 

 the velocity of rotation. Now, according to the demonstration 

 of the preceding theorem, the axis of this couple is perpendicular 

 to the plane loA, and its moment is easily seen to be 



/uajft)^ sin loA ; 



where /x is the moment of inei'tia of the gyroscope, w the velo- 

 city of the earth's rotation, and o), the angular velocity of the 

 instrument. But, inasmuch as this couple maintains the axis 

 of the gyroscope in apparent repose, it is the equal and contrary 

 couple wbich causes the instrument to move. 



The latter is decomposable into two others : the one whose 

 axis is in the plane P will be destroyed ; the other, having its 

 axis perpendicular to P, is alone efficacious, and is represented 



y /iwwy sin loA cos (P, loA) ; 



