341 



I. The component round the axis of figure of the [absolute] angular 

 velocity = Constant = n. This follows directly from Euler's well-known 

 equation for the motion round a principal axis, — 



In the present case, 



A = B iV=0 ,-.^ = 0 r^n. 



at 



Since component of the absolute angular velocity round any line = com- 

 ponent of apparent angular velocity + component of angular velocity of 

 the earth, the apparent angular velocity round the axis of figure 



= ;^ - w cos ^, (1) 



where (^) = angle between axis of figure and polar line. 



II. The equation of relative vis viva, which in this case assumes the 

 simple form. 



2 _ 2 {mv^^) = vo\ (/- /o).« (2) 



* It is at this point that my course and my results differ from those of M. Quet. He 

 writes this equation, S (mv'^') — 2 (mvo^) = 0. To explain the origin of the discrepancy — 

 instead of choosing our co-ordinate axes passing through the centre of the gj^roscope, let us 

 choose them passing through the centre of the earth. The equation of relative vis viva 

 would then be 



- Emv^ - ^mvq^ =2 J Fdp + 2 J 2m F'dp'. 



Where P = force of earth's attraction, F' — centrifugal force due to earth's diurnal rotation. 

 These two forces might be combined for each element into their resultant the force ge- 

 nerally understood when we speak of " gravity," and the last member of the equation might 

 be writtten 2j'SmItdr. Now, in strict accuracy, neither of these forces P andP'is uniform in 

 magnitude and direction throughout the body of the gyroscope, and, therefore, neither of 

 these integrals vanish. Butin seeking to simplify the problem by an assumption sufficiently 

 near the truth, two courses are open to us : — One, that taken by M. Quet to assume the 

 compound force (P) as uniform in magnitude and direction, and that its resultant, accord- 

 ingly, passes through the centre of figure. He thus gets rid of the second member altogether. 

 The other course, which I have followed here, is to treat the earth's attraction only as uni- 

 form, and make no such assumption about the centrifugal force, but to replace 2^'2mFdr by 

 its accurate value, J_ Jq). This hypothesis, the uniformity of the earth's attraction, re- 

 quires only to give it validity that the dimensions of the gyroscope be small compared with 

 the earth ; while M. Quet's assumption requires, in addition, that the earth's angular velo- 

 city be small compared with that of the gyroscope. Now, it seems more logical, in discussing 

 phenomena arising from the earth's rotation, to include all terms springing from that 

 source. The differential equations so found possess this advantage, that they would not 

 cease to hold good were the earth's angular velocity supposed of co-ordinate magnitude 

 with the gyroscope's. Moreover, applying the equations to the case where the axis of the 

 gyroscope is unconstrained, we obtain on this hypothesis an exact solution ; while M. Quet, 

 after an elaborate analysis, has to remain satisfied with an approximation, the simplifying 

 assumption which he made at the beginning precluding him from obtaining a solution in 

 inite terms. 



E. I. A. PEOC. VOL. VIII. 2 Z 



