IN THE NEIGHBOURHOOD OF A CAUSTIC. 599 



±5.6, the largest term in the series is 1 69-044826 : and it is necessary to proceed as far as the 

 45"' power of m. The result +0'000114 for m = - 5'6 is obtained by combining the sum of 

 positive terms + 614-149962 with the sum of negative terms - 614-149848 : and the result 

 + 0'414595 for m = + 5-6 is obtained by combining the sum of positive terms + 6l4'357203 with 

 the sura of negative terms - 6l3'942608. For values of m greater than ± 5'6, the calculation must 

 be made in natural numbers. 



The agreement of the values of the integral, computed by methods so totally different, is not 

 a little remarkable. On the one hand, it may be received by some persons as a proof of the 

 correctness of that part of the theory of the series which asserts the evanescence of the integral of 



a cosine when the limits are and - : on the other hand it may be considered to afford evidence 



of the great care with which the quadrature computations had been made. 



For the last two or three sets of numbers compared, there is a trifling discordance. It will 

 be remarked that in ray account of the computation by quadratures I have shewn that difficulties 

 begin to arise in the accurate computation for the values of m approaching to 4'0, (unless the 

 actual summation were carried to higlier values of w than I carried it in those computations). 

 That the source of the discordances is in these difficulties and the consequent inaccuracy of the 

 quadratures, and not in the inaccuracy of the series, is evident from the following consideration. 

 The numbers computed by the two raethods agree well for the values of m - 4-0, — 3*8, - a"6: and 

 as the quadratures there present no difficulty, it is reasonable to suppose that both sets of numbers 

 are accurate (within such limits as are possible for the sums of numerous figures). Now the terms 

 of the series combined to form the value of the integral for m = + 4'0, + 3-8, + 3-6, are exactly 

 the same as those by which the value of the integral for wi = - 4-0, - 3'8, - 3-6, is formed : the 

 only difference being that they are combined in a diflPerent manner, and therefore, from the evident 

 accuracy of the series for m = - 4-0, — 3-8, — 3'6, we are entitled to infer the accuracy of the 

 series for m = + 4-0, + 3-8, + 3-6. 



G. B. AIRY. 



Royal Observatory, Greenwich, 

 March 24, 1848. 



4h2 



