CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 177 



This mode of determining the constant C is anything but satisfactory. I have endeavoured in 

 vain to deduce the leading term in U for n negative from the integral itself, whether in the 

 original form in which it appears in (5), or in the altered form in which it is obtained from 

 (6). The correctness of the above value of C will however be verified further on. 



10. Expressing n, U in terms of m, W by means of (8) and (9), putting for shortness 



,1 



in\* /m\s 



*- 8 b) =7r U)' < 27 > 



where the numerical values of m and n are supposed to be taken when these quantities are 

 negative, observing that l6^/(3»') = T-l<p, and reducing, we get when m is positive 



Wmih(sm)-ilBcOtU-^\ + SnnU-~\\, . . . (28) 



where 



1.5.7.11 1.5.7.11.13.17.19-23 

 ™ 1 .2(720)" ~ 1 . 2 . 3 . 4 (72 (p)* 



1.5 1 .5.7.11 • 13.17 ( ' 



™ 1 .720 1 . 2.3(720) 3 



When ra is negative, so that W is the integral expressed by writing — m for m in (l), we get 



i , x x a. f !- 5 1.5.7.11 ] 



W=2i(3m)-i e -'t>{l • + ; — --...}. . . .(30) 



v ' \ 1.720 1.2(720) 2 J v ' 



11. Reducing the coefficients of A~ l , (p'"... in the series (29) for numerical calculation, 

 we have, not regarding the signs, 



order i ii iii iv v vi 



logarithm 1-841638; 2-569766; 2-579704; 2-760793; 1.064829; 1-464775 

 coefficient .0694444; .0371335; .0379930; .0576490; .116099; .291592. 



Thus, for m = 3, in which case (p - 7r, we get for the successive terms after the first, which 

 is l, 



.022105, .003762, .001225, .000592, .000379, .000303. 



We thus get for the value of the series in (30), by taking half the last term but one and a 

 quarter of its first difference, . 980816 ; whence for m = 3, W = 6"^ x . 98O8I6 e' v = . 0173038, 

 of which the last figure cannot be trusted. Now the number given by Mr. Airy to 5 decimal 

 places, and calculated from the ascending series and by quadratures separately, is . 01730, so 

 that the correctness of the value of C given by (26) is verified. 

 For m = + 3 we have from (28) 



W= -3-l(R - S) = - 3-*(.9965 -.0213)= -.5632, 



which agrees with Mr. Airy's result -.56322 or - .56323. As m increases, the convergency of 

 the series (29) or (30) increases rapidly. 



Vol. IX. Part I. 23 



