IN GEOMETRY, MECHANICS, AND ASTRONOMY. 427 



Sun in a circle uniformly ; in other words, we shall suppose that w,*, y and r' are constant. Hence, 

 observing that x'= i\y . ii\ (45) becomes 



rfft) '\m' 



^ = 7"5^(^"-7)(/>'"'-7) (*fi)- 



Now let u and \i! be two unit axes at right angles to y and to each other, one of which (a') 

 jioints to the first point of Aries: also let ^" be a unit axis pointing to the north solstice: then, 

 if we assume tit to denote the obliquity of the ecliptic, and tit the Sun's longitude, we evidently 

 have 



/3"= /3'cos ■sr + 7 sin -sr, u'= r' (a' cos ?i't + /3"sin n't), 



and .-. u'= r {a' cos n't + /3'cos -ar sin n't + 7 sin T<r sin n'tl. 

 Hence we have 



Am', y = /sin sr sin n't, 



Du'. 7 = - £>y ■ u = r (rt'cosTj-sin n't - ^'cos n't), 



in 

 and therefore, observing that «'- = -7^, (46) becomes 



— = 3 71 ^X sin TcT )« cos-zr i\wnt — p cos ra tsinnt\ (47). 



By integrating this equation we find w, i.e., w^a + w.fi + w.^y ; and therefore, by equating the 

 coefficients of a, f-i, 7, we find (Oi, w^, 013; from which it appears that oi^ is con'stant (as has been 

 shewn before), and <«, , w^ are small quantities. 



Now. if « denote the numerical magnitude of w, we have 



tl = \/u)i' + ft)/ + ftlj". 



Also the sine of the angle which the axis y makes with the axis ui is 



\/w': 



+ ft) ' + ft),' 



But, since n't varies but little in one revolution of the Earth, it follows from (47). that we may 

 regard ftj, , ftij, W3, as invariable for one day in quantities multiplied by \. 



Hence it follows, that in a day the axis 7 describes a conical surface round the axis w (i.e. the 

 instantaneous axis) with a uniform angular velocity n : and therefore the mean daily motion of the 

 axis 7 must be the same as the motion of the axis uj; or, in other words, observing that the numerical 

 magnitudes of 7 and w are 1 and n respectively, we have, as far as the mean daily motion of 7 is 

 concerned. 



"© , 



dy \nl 1 dw 



dt dt n dt ' 



Hence by (47) we find 



dy an' , , . , 



— = - X SHI TeT ja cos •ar Sin 71 1 - ii sm w t cos re /[ ... (4H). 



dt n 



Which equation completely determines the motion of 7 the Earth's north pohir axis. 



• II in «o cany 10 «ec thai Ihc ciicffic-icnl of y in the firn niumber of ( 4a ) i» ( I + X ) !"" , »">l Ihiit in thi- vccinil njcinhtT ii ii zero . 

 therefore w-^ in conKtant, whether A he ^inial) or not. 



