95 



by the assertion of an improbable event, entirely upon the second. 

 The object of this section is, by a closer examination of the mathe- 

 matical problem of evidence, to ascertain the accordance or non- 

 accordance of the results of usual data with usual notions. The 

 result of the examination is, that common notions, as in other cases, 

 are found closely accordant with theory. For instance, if there be 

 n possible things which can happen, so that the mean probability of 



an event is — , a witness of whom we know no "particular bias towards 



n 

 one mode of error rather than another, asserting an event of which 

 the d, priori probability is a, has his previous credit raised, unaltered, - 



or lowered, according as a— — is positive, nothing, or negative. So 



n 



that though the a priori probabilities were distributed among a mil- 

 lion of possible and distinguishable cases, yet a witness asserting one 

 of them against which it is only 999,999 to 1, would have as good 

 a right to be believed as though there had been but two equally pro- 

 bable cases, of which he had asserted one. 



March 11, 1850. 



On the Numerical Calculation of a class of Definite Integrals and 

 Infinite Series. By Professor Stokes. 



In a paper " On the Intensity of Light in the neighbourhood of a 

 Caustic," printed in the sixth volume of the Cambridge Philosophical 

 Transactions, Mr. Airy, the Astronomer Royal, has been led to con- 

 sider the integral 



W=/ cos— (w 3 — mw)dw, 



and has tabulated it from m= —4 to m= +4 by the method of qua- 

 dratures. In a supplement to the same paper, printed in the fifth 

 part of the eighth volume, Mr. Airy has extended the table as far as 

 m= + 5'6, by means of a series proceeding according to ascending 

 powers of m. This series, though convergent for all values of m, 

 however great, is extremely inconvenient for numerical calculation 

 wben m is large, and moreover gives no information as to the law of 

 the progress of the function for large values of m. The author has 

 obtained the following expression for the calculation of W for large, 

 or even moderately large, positive values of m : 



W=2 (3m)~* JRcos (<p— J Y+S sin (?—■?) j. 



R _ 1 _ 1.5.7.11 1.5.7.11.13.17.19.23 _ 



1.2(72p)« 1.2.3.4(72p) 4 



g _ 1.5 1.5.7.11.13.17 

 1.72<p 1.2.3(72^)3 



When m is negative, and +mw is written for — miv in the integra 



where 



