234 Proceedings of the Royal Irish Academy. 



The centrifugal force in the direction of *S' is, obviously, 



r sin 6 cos 6^ I n + -^ 1 . 



The sixth of equations (2) therefore becomes, remembering that 



,dx' du' d& 



•^ dt dt df 



and equating couples in the meridional plane — 



If we now equate the couples in the equatorial plane, we find, 



since 



dy dz ^ . . ^,[ d4'\ 



z -f - w — = r^ sm- 6' n + -—], 



dt -^ dt \ dt f 



The three equations just found from elementary principles are 

 exact equivalents of the Laplacian differential equations, which are 

 thus expressed by Airy^ : — 



dr r ' r r 2r df r \dt 



dp ^ y- -, Yxy+Zx% ±(^,(f^' 



r^ sin 9' cos 0' [ n + '^Y, (B') 



f'^Zy - Yz- 2r — sm- 6' « + -^ 

 d(fi' ^ dt \ dt 



-2.^.in.cos.'f(,„.f)-,^sin'.f.(C') 



1 " Tides and "Waves," p. 264. 



