Chessin — On the Motion of Gyroscopes. 81 



1. By putting sin 6^ = 0, i. e. at the start the axis of the 

 torus is parallel to the axis (OH) of rotation of the meridian 

 circle (C^). Equation (9)' shows that in this case f^e pre- 

 cession has a constant value, and equation ( 10), that the angular 

 velocity of rotation of the torus about its axis NN' is constant. 



2. By putting ^o = — ^' ^^ this case the precession has 

 the constant value — cd and, besides, '^' = '^o'. This result is 

 a priori obvious. In fact, the assumption of <^o' = — ®> 

 together with 6^' = 0, is equivalent to assuming that at the 

 start the axis of the gyroscope was fixed in absolute space (not 

 relatively to the meridian circle). But we know that, in this 

 case, the axis retains its fixed position in absolute space. 



3. By putting C7j = 6 (w + ^o') cos ^o» ^^* which means 

 the same thing, 



Cyjr.' + iC — b) (<f>o+<o) CO8^, = 0. 



Again, we find that both the precession and the angular 

 velocity of rotation of the torus about its axisy have constant 

 values. 



We now pass to the gyroscope on the surface of the Earth. 



THE GYROSCOPE OF FOUCAULT. 



First of all, let it be remarked that we may neglect in 

 our calculations all terms of the order of co^ and higher, w 

 being the angular velocity of rotation of the earth, if we 

 assume that the force of gravity is constant in magnitude and 

 direction for all positions of the gyroscope.* However, as 

 we gain nothing by thus changing the appearance of our 

 formulas, while we would be obliged, on the other hand, to 

 retain the terms of the order of w^ if the point of observation 

 were near one of the poles, it is advisable to leave the for- 

 mulas as they are, with the explicit understanding, that the 

 results are correct only to terms of the order of(o. 



* See a paper by the author in the Transactions of this Society, Vol. III., 

 No. 1, On the true potential of the force of gravity. 



