INVOLVING THE PEODUCT OP TWO LAPLACE’S COEFFICIENTS. 
583 
The value of f QanQsn'— i dp, when n' is indefinitely great, is therefore identical with the 
coefficient of e ' 2n '~ 1 in the expansion of 
1.3.5 (2n — 1) (I + e' 2 )^ 
2. 4. 6.... 2 n 
Now 
+o d .... + 
1.1.3.... (2n r — 3) 
2.4.6.... 2 7i' 
Jin'-l . 
and by a known theorem, when n' is indefinitely great, 
1.1.3 (2»'-3) 1 1 
2. 4. 6... 2 n 1 ~2n'-l , p n 'i 
Finally, therefore, | Q 2 ,Aj»'-i dp, when n' increases without limit, takes ultimately the 
form 
1.3.5.... (2ft — 1) 1 
2 .4. 6. ..2rc 2-nhf 
If n be very great (though infinitely small, perhaps, compared to n') this becomes 
+■ — .n. In a similar manner it may be proved that the limit of \ Q,„_iQ when 
2 Ttn 2 rv* Jo 
n' increases indefinitely, is 
, 3 . 5.7 ...( 2 » — 1 ) 1 
± 2.4.6....(2n-2) ' 4 *-*n'* ‘ 
If, now, n increases without limit, we obtain 
2i m'i 
There is no inconsistency in the non-agreement of the values found when n and v! are 
indefinitely great, for the limiting circumstances contemplated in the two cases are in 
reality quite different. It may be convenient for the sake of comparison to repeat here 
the equation 
dp— 2 n + 1 ’ 
which is true whether n be great or small. 
The annexed Table contains the exact numerical values of the integrals for which 
neither suffix is greater than eleven. If we fix our attention on a given value of n (say 6), 
while n' varies, we see that the integrals, or, rather, those of them which do not vanish, 
begin by being alternately opposite in sign, and increase in value up to n'—n—l ; that 
when n' — n a change of sign is missed, and that for greater values of n' the regular alter- 
nation of sign is reestablished in conjunction with a steady diminution in numerical 
value. 
This is in accordance with what might have been expected from the general character 
of the functions Q. They have their maximum (arithmetical as well as algebraical) 
value, namely unity, when [a = 1, an even function, Q 2n , vanishing n times, and an odd 
function, Q 2 „ +1 , vanishing n-\- 1 times for values of (m ranging from 0 to 1 inclusive. 
