28 



._, ^ /r ^ (^ — y^) sin vdy cos vS^ 



(6) ,, cos {f — g)= \ '-^ \ 



(7) t ^'^ if -9)- ~^' 



F ^' ''' 1 -_ (1 — y2) sin2 v6^ 

 whence we obtain 



(G y^ (1 - y^) (1 - 2 sin-^ vd,) 



^^ \FJ 1 _(l_y2)sin2^d, " 



we know y is a real quantit}'^ satisfying the conditions 



(9) 1 — y- > and y^ > ; 

 thus the expression 



(10) 1 — (1 — y2)sin2j'(J, 

 cannot be ^ except when 



(11) 7 = and vd^ = ^n. ^n etc. 



Hence, having regard to the first equation (25) on page 31 of Pro- 

 fessor Voigt's treatise, we conclude from (6) and (8) that 



(12) tg2(jp = tg2r(5, 



and this conclusion applies to all allowable values of y and of (Jj 

 with the unessential exception of the case when 



(] 3) 6., = : y = ; vd^ = ^n. f tt etc. 



In the case when d^ = 0. the theorem of course is well known 

 to hold true and may be verified independently. 



§ 3. Before proceeding further it may be well to recall a 

 simple kinematical proposition. As in a previous paper*), let us 

 consider an elliptic vibration 



(la) 5 = a cos « {t — a) 



(lb) r}=^b cos n it — ^) 



due to the superposition of two opposite circular vibrations. To in- 

 vestigate the direction of the revolution in the ellipse observe that 



d /Tj \ nb sin n (a — /Sj ,_ 



dt\^)~~' a~cos^ n {t — a) ' ' ^ 



<) Bulletin Int. tor March 1908. page 129. 



