538 The Rev. S. Haughton on Physical Geology. [Dec. 20, 



The general equations (1) become, therefore, 



Ac?p + (C— A)qr dt=—~Fp sin u du,^ 

 Adq+(A—C)jpr dt= — Ip cos u du, j- .... (6) 

 Cdr =0. J 

 Transforming p, q into p, u, we find, after some reductions, 

 Adp = —~Fp sin 2w dw, -\ 



Ad%=(C - A)rdt— F cos 2w <2t*, L (7) 



Gdr=0. J 



Integrating these equations, and making u, t vanish together, we 

 obtain, as a second approximation to the motion of the pole, 



F sin 2 u 

 P = Po e A > 



u F sin u cos u > (8) 



u=n t — 



J 



These equations show that a secondary wabble of half the period of the 

 primary wabble is superadded to the motion of the pole, which continues 

 to revolve round the axis of figure in 304*75 days, with the component 

 of rotation round that axis always constant ; while the equatorial com- 

 ponent has a periodic variation passing through all its changes in 152*37 

 days ; and the velocity of revolution of the primary wabble is not uniform, 

 but also subject to a periodic variation of 152*37 days, because sin 2 w 

 and sin u cos u pass through all their changes in half a revolution. 



!F 



These results are independent of the magnitude of — , and would hold 



A 



true even if that coefficient were large, which it is not. The motion of 

 the pole would continue for ever, compounded of the primary and 

 secondary wabbles; and in order to obtain the effects of friction in 

 destroying both motions, it becomes necessary to proceed to the third 

 approximation. 



Third Approximation. 



"We must now introduce into the equations of motion the frictional 

 couple acting round the axis 0^', viz. 



F 



~2 (o sin (f> du 2 . 

 Writing for w sin its value p, we find 



Adjp + (C — A)qr dt= — ~Fp sin u du — ^ p du 2 cos u, 



F > (9) 



Adq-\~ (A— C)p*r dts= — ¥p cos u du — -^p du 2 sin u, ( 



Cdr=0. 



