stability of rotating masses of liquid 203 



§ 4. I believe that the discrepancy of conclusion arises in 

 another way. Sir George Darwin computes his form N only to a 

 limited number of terms, and is satisfied with the verification 

 {loc. cit., p. 380) that the terms he would next calculate are very 

 small in comparison. It is easy to show however that the terms of 

 the first few orders, in Mr Jeans' expression for the normal variation 

 from the ellipsoid to the contiguous pear, involve terms extending 

 to infinity in Sir George Darwin's expression for this normal 

 variation. 

 • Taking a point {x, y, z) near to a point (xq, jJq, Zq) of the ellipsoid 



a;> + i/lh + s2/c = 1, 

 given by 



Darwin {loc. cit., p. 320) has an expansion in Lame products 

 - lA/V = const. - eQjo'^ "A^io -A'^'^oi - W^~^ 



where Pq is the central perpendicular on the tangent plane at 

 {xq, yQ, Zq), so that, in terms of elliptic coordinates, Pq^ = abcjfiv. 



Jeans' form of contiguous surface {Phil. Trans. A, ccxv, ccxvii, 

 1915, 1916, or Costnogony, p. 88) is 



/y*^ ^j^ ly^ 



0= — l^ f-'-T-H h exu. + cHu + e^xv^ + 



a c 



where %, ^2? ^2 ^^e integral polynomials in cc^, if', z^, 1, of respective 

 orders 1, 2, 2. If herein we substitute 



.^a + A\* A , A A2 





with similar expressions for y and z, and solve for A, we shall find 

 a series of the form 



— 2 = — CXqU + C^PqH + C^XqPq^IV + ... , 



Po 

 where u, v, w, ..., are polynomials in Xq^, ^q", Zq^, which, on ex- 

 amination, prove not divisible by — ^ or Xq^/ci'^ + yo^/b^ + ^qI^^- 



These are then expressible by finite series in Lame functions. But 



nbc ct u c 



p^ = ' - ^ Pq4 ^ — __^ ^ can only be expressed by infinite series 



of this form. 



Thus even the terms of the second order in Mr Jeans' form for 

 Xjpf^ cannot be put down exactly from Darwin's results without 

 taking account of the whole aggregate of terms in Darwin's series; 

 and the same for terms of higher order. 



