On the Diurnal Changes of the Atmosphere. 427 



nate functions of the modulus; notwithstanding that they are 

 given by the apparently determinate conditions^ _ 



_i_ ■'■ n; ~ '' 



I /*" dx 



-f 







In fact definite integrals are in many cases really indeter- 

 minate, and acquire different values according as we consider 

 the variable to pass through real values, or through imaginary 

 ones. Where the limits are real, it is tacitly supposed that 

 'tthe variable passes through a succession of real values, and 

 : thus CO, u may be considered as completely determined by 

 these equations, but only in consequence of this tacit suppo- 

 sition. If c and e are imaginary, there is absolutely no system 

 of values to be selected lor w, u in preference to any other 

 system. The only remaining difficulty is to show from the 

 integral itself, independently of the theory of elliptic functions, 

 that such integrals contain an indeterminateness of two arbi- 

 > trary integers ; and this difficulty is equally great in the sim- 

 plest cases. Why, d p7iori, do the functionSii v <v ifoiVjuiji 

 . , r dx , "■' pdh -" "f " 



contain a single indeterminate integer ? 



Ohs. I am of course aware, that in treating of the proper- 

 ties of such products as 11 I 1 + J, it is absolutely 



^ \ mw ■\- 7iv/ 



necessary to pay attention to the relations between the infi- 

 nite limiting values of w and w, and that this introduces cer- 

 ' tain exponential factors, to which no allusion has been made. 

 But these factors always disappear from the quotient of two 

 such products, and to have made mention of them would only 

 have been embarrassing the demonstration without necessity. 



LXVIII. On the Diunial Cha?iges of the Aqueous Portion of 

 the Atmosphere^ and their Effects on the Barometer. By 

 Thomas Hopkins, Esf.*»^ — ^ 5^- — 'c 



T is admitted by meteorologists, that the various quantities 

 of aqueous vapour which exist in the atmosphere during 

 the different hours of the day, contribute to the production of 



• Communicated by the Author ; having been read at the Manchester 

 Literary and Pliilosophical Society, October 7, 1845. ' '*" ^*- " 



2 F 2 



I 



