Be ssel- Clifford Function, and its applications. 517 



Then the D.E.'s for M antl p become 



,cim , . . ,dM 15,, . 



*(l-*)- 5 ^- + (-i-2ar)- 3S +~M = 0, . .. (9) 



dN 

 with -=- = M and (11) changed into (10) by a change 



of a* into 1 — # ; and these H.G.D.E.'s have an algebraical 

 solution, x% for M, xi for N, and (1— ,r)f for p. 

 Th.9 pressure equation becomes 



dp = — M/o --2 = — §7r/9 c 2 (l — x) 2 dx, 



p = **p*?o--*y = ***?(£$, ■ ■ (12) 



\po/ 

 in agreement with Schuster's results. 



But in rotation, rj and e become infinite at r = 0, so that 

 this conglomeration is unstable at the core. 



This suggests a generalisation, of putting 



4?rr 4 dp _ _4n(n~f l)_c-V 

 M rfr ~ (c 8 + r 8 ) 8 ' ' ' * * ( ld ) 



replacing ^ in these equations by n(n + l), and then the 

 H.G. D.E.'s have algebraical solutions. 



With 2x = l±t, 2(l-x) = l + t, p or N=y, (10) and (11) 

 assume the same form 



(l-^)|f + 3| + »(n + l,) y = 0: . . (14) 



But the integration of the pressure equation does not lead 

 to any simple relation between pressure p and density p. 

 corresponding to a physical law, except in Schuster's case of 



2' 



The change from M to N in ('.)) and (11) suggests that, 

 in the general H.G.D.E. for y. 



i(l~^||-H[7-C« + /3+J)a;]^~ a ^==0, . (15) 



a differentiation leads to the H.G.D.E. Cor "• = - , ' / , in which 

 a, /3, 7 are changed into a + 1, /3 + 1, 74 1, 



Phil. Mag. S. 0. Vol. 38. No. 227. Nov. 1919. 2 O 



