VIII. On the numerical Calculation of a Class of Definite Integrals and Infinite 

 Series. By G. G. Stokes, M.A., Fellow of Pembroke College, and Lucasian 

 Professor of Mathematics in the University of Cambridge. 



[Read March U, 1850.] 



In a paper " On the Intensity of Light in the neighbourhood of a Caustic*," Mr. Airy the 

 Astronomer Royal has shown that the undulatory theory leads to an expression for the illumi- 



nation involving the square of the definite integral / cos — (w> 3 — m w) d w, where m is pro- 



•'o 2 



portional to the perpendicular distance of the point considered from the caustic, and is reckoned 

 positive towards the illuminated side. Mr. Airy has also given a table of the numerical values 

 of the above integral extending from m = — 4> to m = + 4, at intervals of 0.2, which was cal- 

 culated by the method of quadratures. In a Supplement to the same paper -j- the table has 

 been re-calculated by means of a series according to ascending powers of m, and extended to 

 m = ± 5.6. The series is convergent for all values of m, however great, but when m is at all 

 large the calculation becomes exceedingly laborious. Thus, for the latter part of the table 

 Mr. Airy was obliged to employ 10-figure logarithms, and even these were not sufficient for 

 carrying the table further. Yet this table gives only the first two roots of the equation W= 0, 

 W denoting the definite integral, which answer to the theoretical places of the first two dark 

 bands in a system of spurious rainbows, whereas Professor Miller was able to observe 30 of 

 these bands. To attempt the computation of 30 roots of the equation W = by means of the 

 ascending series would be quite out of the question, on account of the enormous length to 

 which the numerical calculation would run. 



After many trials I at last succeeded in putting Mr. Airy's integral under a form from 

 which its numerical value can be calculated with extreme facility when m is large, whether 

 positive or negative, or even moderately large. Moreover the form of the expression points 

 out, without any numerical calculation, the law of the progress of the function when m is 

 large. It is very easy to deduce from this expression a formula which gives the i th root of 

 the equation W = with hardly any numerical calculation, except what arises from merely 



(Tn\ i 

 passing from I — J , the quantity given immediately, to m itself. 



The ascending series in which W may be developed belongs to a class of series which are 

 of constant occurrence in physical questions. These series, like the expansions of e~*, sin at, 

 cos a?, are convergent for all values of the variable at, however great, and are easily calculated 

 numerically when <r is small, but are extremely inconvenient for calculation when a; is large, 



• Camb. Phil. Trans. Vol. vi. p. 379. t Vol. vm. p. 595. 



