•518 Sir G. Greenhill on the 



Thus, as in (5) § 1, the D.E. can be replaced by a sequence 

 relation. 



An attempt has been made by G. W. Hill in the Acta 

 mathematica on the assumption of a linear law, p = A/>+ B, 

 connecting density and pressure. And other memoirs may 

 be consulted : Comptes Rendus, April 13, 1885, Radau 

 " Sur la loi des densites a l'interieur de la terre." Bulletin 

 ■astronomique, 1884, Tisserand and Stieltjes. 



13. The pressure equation in (25) § 11 does not appear 

 tractable on the theorems given by Lommel, Math. Ann. xiv., 

 for the integral or! the product of two Bessel Functions. 



Changing to the Clifford Function, and adopting Lommel's 

 method, 



~^G n G n =p 1 C- 1 G m G n -xP(C m+1 (\ + G m G n+i ), . (1) 



— x v (C ni+ 2Cn + i + C n+1 C, l+2 ) } (2) 

 and then, writing equation (5) § 1 as a sequence equation, 



*a +2 = (n + 1)0.4.1 -C», ... (3) 



equation (2) becomes 



J^+i Cw+1 C n+1 = (p + l)^C w+1 C re+1 



-xP(m + n + 2)C m+1 C n+l 



+ ^(C m +iCV4-C, n C w+1 ), . . (4) 

 and adding (1) and (4), 



= pxP- l G m C n r (p — m -n—1) G w+1 C n+ i. (5) 

 Thus with p=0, 



(m + n + 1) J G m +iC n+l d.v = — xC m+1 O n+1 — G m C n , (6) 

 and with p = m + n + l, 

 {m + n + l)\ x m + n G m O n dx = x m + n +\M m+1 G n+1 + C m ;i ) ; (7) 



easily verified by differentiation. 



But the pressure equation (25) § 11 cannot be integrated 

 by means of these relations, except for n = 2. 



