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Full text of "Scientific Papers - Vi"

1916]                    WHENT  n.  IS  GREAT  AND   6 HAS  ANF VALUE
so that
v     2.4.6...     n
Here n is even, say 2r, and it is supposed to be great.    Thus
22.42. 6a
and when r is great,
8r ' 128r2
When n is even and with this value of G,
When , is odd, the same value of 7, viz. -?r, secures the required evanescence in (8), and we may conjecture that the same value of C will also serve. Laplace* indeed was content to determine 7 from the case of n odd and G from the case of n even. I suppose it was this procedure that Todhunterf regarded as unsatisfactory. At any rate there is no difficulty in verifying that (9) is satisfied by the same value of C. From that equation and (10),
1.3.5...      n
(n + 1) TT
1     4(n + l)
32~(?i + I)2 """ 128 (n + 1)
Here, as throughout, m  n+^, and when we expand these expressions in descending powers of n we recover (11). Equations (11) and (12) are thus applicable to odd as well as to even values of n.
* M6c. C61. Supplement au V volume, f- Functions of Laplace, etc. p. 71.