Equilibrium of Rotating Fluid. 677 



Both curves plainly asymptote the z axis, and when z is 

 large the curve (9) is below the curve (8) unless n = l } for 

 when z is large the equations are 



and 1 Ct>2j 1 



"Jo (2n + l> 



Moreover, both curves continually descend from £ = 

 to oo , and both are convex to the axis of z, for in both 



-/- is negative, and -j4 2 positive, for all positive values of z. 

 dy d 2 u 



Since, further, both j ; and ~ are continuous functions 



of z from z = to go , it follows that there cannot be more 

 than one real solution of equation (7) between and co . 

 Again, when 2=0, y=l on curve (8), and on curve (9) y is 

 infinite if n is even, but is unity if n is odd, for 



= 1 — 1 ~P-2n-] x (a rational integral expression 



•J- 1 of degree 2n — 3)d,r 



= 1. 



Hence there is one and only one root of equation (6) if n 

 is even, and no root if n is odd, unless n = l, in which case it 

 will be found that (7) is an identity. 



The proof further shows that two of the curves (9), for 

 different values of ??, cannot intersect. For when z is small 



the difference of the ordinates is 1 ( PL, — P 2 ) ' , which 



Jo V *'*" 



is positive (infinite) if n is greater than m ; and when z is 



large the difference is (- -_- -)- which also is 



positive. \2m + l 2n + \J z 



We have thus proved that that member of the system of 



