XLI* Supplement to a Paper " On the Intensity of Light in the neighbourhood of a 

 Caustic" By George Biddell Airy, Esq., Astronomer Royal. 



[Read May 8, 1848.] 



In a Paper "On the Intensity of Light in the neighbourhood of a Caustic" communicated to 

 the Cambridge Philosophical Society about ten years ago, and printed in the 6th Volume of their 

 Transactions, I shewed that the expression for the intensity of light near a caustic would depend 

 on the infinite integral 



f 



Jw 



cos — (zi) ' 

 2 



m.w)'* 



from ai = to to = 



where to is a quantity proportional to the distance of a point from the geometrical caustic, measured 

 in a direction perpendicular to the caustic, and estimated positive towards the bright side of the 

 caustic : and I gave a detailed account of the method of quadratures by which I had computed the 

 numerical value of this infinite integral for the values of m - 4-0, - 3'8, &c. as far as + 4-0 ; and 

 I exhibited in a table the computed values of the integral. 



The computation by quadratures was exceedingly laborious, and I did not resort to it without 

 trying other methods of a more refined nature. But in every attempt at expansion of the formula 

 I was met by the integral of a sine or cosine with infinite limits. The reasonings upon which 

 several mathematicians have attempted to establish the value of such an integral appeared to me so 

 little conclusive, that I preferred at once to abandon the expansions which introduced them, and 

 to rely only on the infallible but laborious method of quadratures. 



On my stating to Professor De Morgan, after terminating the calculations, the scruples which 

 had led me to reject the expansions, he expressed himself so strongly confident of the correctness 

 of the conclusions upon the point which I had considered doubtful, that I was induced to undertake 

 the numerical computation of the series given by expansion of the formula. I proceeded at once as 

 far as it was possible to go with 7-figure logarithms, when I was interrupted, and the computations 

 were laid aside for some years. I have lately taken them up again, and have completed them as 

 far as they can be carried with 10-figure logarithms. It is the result of this calculation, and the 

 comparison of this result with that formerly obtained from quadratures, tiiat I now beg leave to 

 present to this Society. 



Before entering upon the numerical investigations, I will transcribe a letter which Professor 

 De Morgan at my request has written to me, and which he has permitted mo to publish. It 

 contains an explanation of his views upon the evidence for the numerical certainty of the results 

 obtained by such integrals as those to which I have alluded. 



• [ retain thi« notation in preference to tliat which is commonly 

 employed, partly becauKC it is familiar to me, and because 1 have 

 u«cd it in the paper to which I refer, partly hecaune I thiiil< that 



any notation which requires the expression of a differential at the 

 end is tor that reason ohjcctionuble. 



