THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 31 



Let -r-be used to denote geometrical differentiation with respect tox i.e., differen- 

 dx 



o 



tiation in space keeping y and z constant and let ^- be used to denote algebraic 



differentiation, i.e., partial differentiation with respect to* keeping y, z, and X constant. 

 Then 



d_ = _3_ + ^X JL &c ( 8 ) 



dx Sx dx 3x' 

 and we have 



dw aw , d\' aw 



dx' 9z' dx' I3X'| 



If the point x'. y', z' is on the boundary we have X' = 0, so that from equation (7), 

 -57 = 0, and it appears that equations (3) will all be satisfied if 



at all points on the boundary X' = 0. 



We notice from a comparison of equations (6) and (7) that 



$ ( X ', y', 2 ', X') d\. = V t (d\.)-V (dX.), 



and the right hand of course vanishes when x', y' , z' is on the shell d\ s . Thus we 

 must have, at all points, 



*(x',?/,z',\') = ......... (11) 



identically, provided that X' has the value appropriate to the point x', //, z'. This 

 condition of course imposes more restriction on the value of $ than does equation (10). 

 Equation (10) was adequate to ensure that the boundary condition (3) should be 

 satisfied, but the remark just made shows that for (3) and (2) both to be satisfied 

 i.e., for W to give the true value of V; V , equation (ll) must necessarily be 

 satisfied. We shall now assume that equation (ll) is satisfied, and proceed to satisfy 

 the remaining condition expressed by equation (2), namely, 



(12) 

 In virtue of equation (ll), equation (9) reduces to 



dx' 



