22 
ME. W. SPOTTISWOODE ON AN EXTENDED POEM OF 
and for the even indices, 
da 2 ”” Vld^ , 
«4 = 5(A3 — (W— 2)«,) 
“"4v^3 a 2.1 ®V’ 
1/a ^"4 (re-4)(n-2) (n-4)(?i-2)n 
~6V ® 4 -^3-1- 4.3 -^1 4.2.1 
But it is better to make use of the operative symbol for the solution of the equations ; 
thus : 
{n--i-{-2)ai_^ A,._i 
|(^^— ^4-2)s“^^4■*|<^^i=A^_l, or |(w— ^+2)^“^*i+l|^’®i=Ai-l j 
whence 
■ j Ai_,, 
^i=y|l+ (^“'^+2)s Af_i, 
If A n— ? + 2^ , (n— ^+2)(/^-■^+4) ^ ] 
i-2 ^*--3+ ■(*_3)(i_4) Ai_5 ..j; 
the last term being 
and 
, ,tA(n— i + 2)(« — ? + 4).. (n~l) , • • jj 
(”) (iX: 2 )(i- 4 ):.i Ao when i is odd, 
, Y{n—i + 2){n — i + 4!)..n , 
(-)^ (f- 3 )(^- 4 ).. 3 — «« when ^ is even. 
The two cases, however, of n even and n odd require special notice. If n be odd, the 
last equation of the series for determining the as may be thus written : 
-_AA 1a I 1.3 . . \-^ 1 . 3 . . /I 
'wa„+,_u— A„— — £A„_2-P A„_4— . — j 2 
which determines 
But in the case of n being even, 
%a„+i = 0=A„- 
1 . 
A-l^'*-2 +(ra-l){ra- 3)"^”-^ ••( )^Ao, 
a relation between the As alone. The explanation of this apparent anomaly is to be 
sought in the complementary arbitrary function arising out of the operation V/'O- 
