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The President having taken the Chair, 



The Rev. Charles" Graves read a paper by Mr. George 

 Boole, of Lincoln, containing investigations supplementary 

 to his former papers on Discontinuous Functions and Definite 

 Multiple Integrals. 



The author commences his observations by pointing out 

 a distinction among integrals which constitute the limits of 

 more general forms, according as they are supposed to be 

 obtained by the vanishing of one or of more constants. The 



integral V dx cos {qx) a;"~\ n being positive, considered as 

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the limit of \ dxa-''^ cos (qx) rc"~', he designates a limiting 



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integral of the first class, because it involves the consideration 

 of one vanishing constant. The same integral, when n is 

 negative, he regards as the limit of the more general form. 



f»* dxz~^'-^ cos qx cos {n tan~' ar) 



two constants k and k' vanishing'; and designates it a limiting 

 integral of the second class. Under this assumption he 

 assigns as its value 



r** dx cos qx _ tt ( ± 5')"'"' 



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