THE EQUILIBRIUM OF FLUIDS. 137 



shown that the value of Y' (i \ when expressed in the new co- 

 ordinates, will be of the form F/ (i) ; but all functions of this 

 form (Vide Mec. Gel. Liv. iii.) may be expanded in a finite 

 series containing 2i+ I terms, of which the first is independent 

 of the angle -cr/, and each of the others has for a factor a sine or 

 cosine of some entire multiple of this same angle. Hence, the 

 integration relative to OT/ will cause all the last mentioned terms 

 to vanish, and we shall only have to attend to the first here. 

 But this term is known to be of the form 



T ( n i.il ,. i.i 1 .i 2 .i 3 ,. . p \ 



H*-2^^ 1 ft| - 1 + 2 .4. 2 f-i.af-8 lV ~ r 



and consequently, there will result 



f 2 ^ >V'd) a if H i'i- 1 /.-a , *.*'-l.*-2.4"-3 ,, 4 p \ 



J ^ F =27r H^ 2^?=I* + 2.4.24-1.27^ - &C 'j' 



where /*/ = cos #/ and A; is a quantity independent of 6 t ' and OT/, 

 but which may contain the co-ordinates 0, -or, that serve to define 

 the position of the axis X v passing through the point p. 



It now only remains to find the value of the quantity k, 

 which may be done by making #/ = 0, for then the line r coin- 

 cides with the axis X v and Y (i} during the integration remains 

 constantly equal to Y (i) , the value of the density at this axis. 

 Thus we have 



rii\ > 7 A, "." 1 ,4.4 1.42.4 3 p \ 



- =2^(1-^^+ 2-4 . 2 ._ 1-2 ._ 3 -&C.J: 



or, by summing the series within the parenthesis, and supplying 

 the common factor 2?r, 



"1.3.5 ......... 2*-! 



and, by substituting the value of Jc, drawn from this equation in 

 the value of the required integral given above, we ultimately 

 obtain 



