-3- W 



:n«t is 



; ♦ Z s„S^ = t 2-' (1 - 2a-' 1 i„S„) * a-^^ I (n * I) 8„S^ («) 



on retaining the terns as far as tne first oroer only. 



Since (8) is true over a eanplete spnere, ■€ nave on equatifig c&sf'icients o' the 

 corresponding nenaonics 



; = ,-^ »; B„ = - 2a-5 » 5„ * {n ♦ 1) '-"^ S^ (») 



or, siVvir^ for t sna 9^, 



A = 'Jk : 3^ ' (j"*' B„* 2>.*- a B„)/(n* I) (lO) 



The rery.ininj oounflarjf condition is that tne pressure is continuous across the surface of tne 

 OuOde. This rsfljires thit the values of o as given By (3) and (7) snail De eaual "tien r = ». 

 SuSstituting froff (3) and U) in (t), me jsiia (lO), we find, te t^< f:"^? cnser, 



{f(a,/a)^- ;}/p.=aS, = .V,5-2 



- I T-r-^ '- U - < V :. - J^ ;,, * a V„ > S^ (11) 



Tfie i:/."'ere'. t ;a^ £j-.3t-.c-.i /: •■ z't. zrz. :^'tJ 



Sirie the spherical haraonic ex|nnsion on the right of (ll) is tf>e expansior of a ccosta';*. 

 over i corvlsts sphere, ^ i^ve 



, V - i a^ = :» '-/'' - a^) /p '7^. (12) 



A ■:. • tl. ' 2=a (13) 



and (1 - n) ' 0^ ♦ >5 ;^ • !! =^ = 0, n ^ 2. (i^ 



Equation (12) agrees "itn the known result •ren -.r^ S'^:ole is accurately spherical, ano it 

 ■^f i-i solved to give a(t) ir terTs of ', 3, p and y. S^stitutinc this valie in (l«), me ootain 

 the aifc'e"*. iil eoi^tion to deteraine o„(*)' Equation (ij) integrates i»<eai2te1)f 



" -- '''' 3 r =if=' <'-'='!>^ , . 



0, = J: ' ^ I a-'ot ♦ i -i , at (i»a) 



Jo Jo -'o 



Tne first part of this expression is Herring's fon»jl3 (») for the rise of tne Suocle due to gravity; 

 tre second Kn represents the effect of an initial velocity. 



Except 'or tne case when 9=0, the solution of (l2) aakes aft) a periodic function of t, i.e. 

 the ojODle Pulsates. icns£3i»ntly tne eqjstion (it) for a is one in itfiich the coefficients are 

 ■jeriodie flections, tne period (T, say) Peir^ equal tc that of the pulsation of the DuBBIe. Prpr 

 'loquet's theory ;f sjcn aquations, it is worn ttet tre general solution is ordinarily of the *,rr 



0, = :>* f(t; * CjS-^* ft-t) (15) 



• •e'e Ij inc -J a''e atitrar, coistnts, and f(t) is a pir\y3'iz 'jrcticn of oerioo T. ■'r* oo'ata-:' 



X and the fjoctio^ '',:) are cete'-i'eo oy :-e periodic coe'f ici='-:s of tne Ki.ation. 



•■en, "o^er, tr* equation pcssesses a periodic solution, say g(t) (correspcndino tc 

 \ = e) tr« general soiut: n is not of the for» (l5); the seccnd solution g2{t) is not periodic Out 

 ^tisfies a functional equation cf tne type 



3j(t * T) = ^^(t) * T 5,(t). 

 *" ix=r!cle :f tnis ."COjrs ir tre oresert c^lCjlitior-^ 'zr ire ojse - = :. 



