HOMOGENEOUS FLUID AT LIBERTY. 509 



attracted point in a level surface, put 6 and ^' for the angles which r and s make with 

 the axis of rotation ; and vx and m' for the angles which determine the positions of 

 the projections of r and s upon a plane passing through the centre of gravity perpen- 

 dicular to the same axis : then -v^ being the angle between the two lines r and *, and 

 f the distance oi dm from the attracted point, we shall have 



y = cos "4/ = COS d COS ^' -|- siu d sin ^ cos (or — zs'), 



f = ^ s^ — 1 sr .y -\- r^. 



Again, if the plane of the two lines r and s describe the small angle <f o- by revolving 

 about r, the extremity of s will describe the short line s cos -^ da perpendicular to the 

 revolving plane : further, supposing that the arc -^ increases to -4/ + <Z -4/, the extremity 

 of s will move through the short line s d-^m the plane of the arc -^ ; now the short 

 lines s cos '<\) d a and s d -^ being perpendicular to one another and to s, the molecule 

 d m may be considered equal to s cos -^ dc X s d-^ X ds\ or, which is the same 

 thing, we may assume 



dm =■ — d y d ff . s^ d s. 



By substituting the values of dm andy, the integral under consideration will be thus 

 expressed : 



the integrations being extended to all values from y= l,fl' = 0, toy= — l,<r = 2^, 

 and from s = r', to * = R', r' and R' being two radii in the same straight line, the first 

 of a level surface, and the other of the upper surface of the fluid. The radical quantity 



must now be expanded in a series of the powers of — , viz. 

 the coefficients being determined by the formula 



2' -[ .2.3. ..idy' 



and having substituted this series, and effected the integrations with respect to d s 

 between the assigned limits, the result will be 



f^-^ff-dyd.i^^-^^) 



+ rf/ 



dyd<r{R'-r')C'^'^ 



+ r^//-dydalogyX C 



* Vide Appendix. 

 MDCCCXXXIV. 3 U 



