86 A. Mukhopadliydy — On Poisson^s Integral [Feb. 



the differential equation holds for every point of every curve of the 

 system^ the first primitive holds only for every point of the same curve, 

 the different curves of the family being obtained by the variation of the 

 constant which occurs in the first primitive. There can be no doubt 

 that, failing to notice this distinction, Ool. Cunningham has given an 

 interpretation which belongs not to the Mongian equation but to one of 

 its first primitives. Indeed, gentlemen, the error into which he has 

 fallen reminds me of an old story with which you are familiar in eastern 

 lore ; you have often been told how an oriental king, desirious of testing 

 the powers of an astrologer placed a fi.nger-ring set with precious stones 

 in a casket, and having closed it asked the astrologer to divine its con- 

 tents ; the astrologer moved the admiration of the prince when after 

 long calculation, he pronounced the contents to be a hard stone, 

 circular in form, with a hole in the centre and an object of every-day 

 use ; but, imagine the disappointment of the king when on pressing 

 the astrologer, he was told that the hidden object was a grinding 

 stone such as you find in every Hindu household. Such is the facility 

 of error where we have to fix upon a particular object from a not 

 very definite description of it, and. Col. Cunningham's paper shews 

 that such an error is possible even in an exact science like mathe- 

 matics. To sum up : the Colonel's remarks do not refer to any of the 

 vital points of my paper, and, as to his geometrical interpretation of the 

 Mongian equation, it is wholly irrelevant. I, therefore, stick to my 

 statement that the true interpretation of the Mongian equation has yet 

 to be found.* 



The following paper was read — 



On Foissons Integral. — By Babu Asutosh Mukhopadhyat, M. A., 

 F. E. A. S,, F. R. S. E. 



(Abstract.) 



The object of the author in the present paper has been mainly to 

 discuss a remarkable definite integral, which was first considered by 

 Poisson in his memoir on definite integrals, inserted in the tenth volume 

 (seventeenth cahier) of the Journal de Vecole PolytecJmique. The 

 paper is divided into four sections, of which the first is introductory. 

 The second section is devoted to a consideration of the transformation 

 of the integral ; the method of reduction is first applied to a generalized 

 indefinite form of Poisson's integral, from which Poisson's result is de- 

 duced with ease ; and the process at once leads to four remarkable 

 definite integrals which are believed to be new. The third section gives 



* Sinde these remarks wore made, I have snccocclcd in discovering the geo- 

 metrical moaning of the diii'erontial eqaation of all parabolas, which I hope to 

 commmiicate to the Society at an early date. 



