Prof. Challis on a Mathematical Theory of Tides. 25 



by integrating that the equation is satisfied if 



S?n/na _ j j , \ 1 



AUIIILLU, , X , *• 



= q^ 3 > and p( r ) = — ~2 ( fl + ^ r ) + Ce "^ 



C being the arbitrary constant introduced by the integration. 

 Consequently 



- Q]p- ( — t Ce a ) cos X Sln HO— fit). 



By means of this value of u' the values of v' and w/ may be ob- 

 tained from the equations 



n du' 3mr*fi rfV 



0=— Ky-jg H g^- cos 2 A, cos 2(0 — fit)—rcos\ — , 



° = - G ^--R^ sin2X S m2(^-^)^^ 

 It will hence be found that 



, Sm^ia/r , 2a 2a 2 2aC 9 {\ 



, 3^a/r 2a 2a 2 2aC ?\ . rt . 



" = 2GR3(g + iVy * J sm2Xsm2(^-^). 



These values of w', «', and w' satisfy the condition of making 

 u'dr + v'rcosXdO + w'rdX an exact differential (dfi); whence by 



integration 



.2 o^v. o„2 o„n £!!> 



, 3™ M tf /r 2 2«r , 2a 2 2«C £\ a . 



* = -<m?\-j + T + Y~T / cos Xsin2 ^-^). 



It is now to be observed that although the form of the func- 

 tion <f>(r), and the above expressions for u', v', w 1 , have been 

 found by satisfying the circumstances of the motion and pres- 

 sure at the limits of the fluid mass, the same forms are applicable 

 at any point of the interior. The reason for this assertion is, 

 that as the motion at each point is determined by the given ar- 

 bitrary conditions, expressions applying to the motion generally 

 can only be obtained by satisfying these conditions. Or it may 

 be said that it is only as being generally applicable that any ex- 

 pressions are capable of satisfying the given conditions. The 

 principle here stated has been admitted by Poisson in his solu- 



