concerning Numbers. 37 



two constants, whose values are assigned by experiment, and 

 which can have no natural connexion with the subject; and 

 as these constants are merely calculated so as to give as little 

 error as possible for the small limits within which the obser- 

 vations are confined, there is no a priori reason for supposing 

 that their assigned values remain permanent for numbers of 

 an order of magnitude far exceeding the observed limit. 



The primary object of the following investigation is the 

 discovery of the laws regulating the occurrence of prime num- 

 bers by means of analysis; and it is evident that if this object 

 can be effected by the deduction of a simple analytical law, 

 we shall be in a position to deduce many curious and interest- 

 ing results connected with the subject. 



The power of applying analysis to this subject is based upon 

 a principle, which at present it would be presumptuous to 

 rank amongst recognized forms of mathematical reasoning; 

 to wit, that the real analytical equivalent of the different 

 values of an indeterminate expression is the arithmetical mean 

 of those different values. It is satisfactory, however, to be 

 able to observe that this dogma is not propounded here for 

 the first time; but has appeared, at least as matter of induction, 

 under the high authority of Professor De Morgan. (Camb. 

 Phil. Trans., vol. viii. part 2. No. 15.) The present is not a 

 convenient opportunity for the discussion of this principle ; but 

 as it is likely to be introduced in some shape or other into 

 analysis, it is very necessary that the grounds upon which it 

 rests, and the subjects to which it may be lawfully applied, 

 and the manner of its application, should be analysed, and 

 recognized to a due extent. I hope on another occasion to 

 be able to advance some reasons for the conclusion, that this 

 principle may be introduced into mathematics without de- 

 parting from or unduly extending doctrines heretofore ad- 

 mitted. At present, I shall merely observe that I do not 

 regard it as probable that the principle in question, or any prin- 

 ciple analogous to it, is deducible from the fundamental axioms 

 of algebra ; that it may nevertheless be true in some definable 

 and useful sense; that, so far as experience leads us, no in- 

 congruity arises from the application of this doctrine, but that, 

 on the other hand, whenever an analytical equivalent is de- 

 duced naturally, it is found to coincide with the arithmetical 

 mean; and lastly, that the subject of this paper affords a 

 favourable case for its application, inasmuch as the results 

 which we propose to obtain are precisely of that average cha- 

 racter of which results flowing from such a principle might be 

 expected to partake. 



In the course of this paper I shall have occasion to employ 



