116 
I)R. G. R. GOLDSBROUGH OX THE INFLUENCE OF 
different value of c on substituting in equation (18), and different values for A and X. 
Hence there are four distinct solutions and these when multiplied by arbitrary 
constants will give the complete primitive of equations (6). 
(b) The Particular Integral. 
We have now to determine the particular integral of equations (4). We shall assume 
only one general term on the right-hand side and take the complete solution as the 
sum of a series of the corresponding solutions. The equations may therefore be 
written 
/j" — 2 kct' + /j- 0] cos rep + cr—61, r sin rep = |-0 3 
<t" + 2 Kf> + /A0 4 r sin rep + oA 0 5 , cos rep = 0. 
Assume 
P — e a, 
and 
rr — C A, 
where X and A as before are functions of ep. On substituting in equations (19) and 
reducing, we find 
—m 2 A+2nnA / +A" — 2/c (cm'X. + X') + A20, ir cos rep + XSQ 2 ,r sin rep = 
—wi 2 X + 2 miXf + X" + 2 k (cm A + A') + A^0 4 ,sin rep + XS0 5j r cos rep = 0. 
As a solution we now take 
A = A o + 2XA r , s e,.. s + 2XXXB r , s , p , ? e r , 5 0 p , 2 + 
x = x 0 +^x r .A. (t +^^Y r , s . p>9 e r . s e / „ ? +.... 
In these summations all the 0’s in the coefficients of p and a are to be included 
except 0, 0 and 0 5iO . A 0 and X 0 are constants, and the other coefficients functions 
of ep. 
Now substitute these expansions for A and X in (20), and equate to zero the terms 
involving no 0 except 0 ] O and 0 5 O . We then have 
—m 2 A 0 —2/fimX u + 0 liU A o = T0 3>J)I , 
— ?n"X u -i- 2/umA(, + 0 5j 0 X Li = U. 
Whence 
A 0 = i0 3 . m (05,o-^ 2 ) 6- {(0 IiO -?n 2 ) (0 5 ,o— m 2 ) —4/Pm 2 }, 
X u = —KimG 3lll -T- {(0^o —m 2 ) (0 5jU -m 2 )-4/c 2 m 2 }. 
Next, taking the coefficient of 0 Lr , we have the equations 
on A hr + 2tmA r + A 4 r — 2/c (i?)iXj ir + X , lir ) + 0j i(l A], o + A n cos r<p = 0 
on Xj_ r -t- 2unX l r + X i+2/c (jttiAj r + A / j !r ) + 0 5 0 Xj r = 0 
* The use of in here to represent an integer is to be carefully distinguished from its previous use to 
represent mass. 
