NOTICES AND ABSTRACTS 



MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



On Diverging Infinite Series. By Professor Young. 



The doctrine of diverging infinite series is a subject upon which very conflicting 

 views are at present entertained. Cauchy, Poisson, and the modern Frenclr analysts 

 generally, characterize such series as false developmentis, and reject tliem accordingly; 

 whilst some of the most distinguished writers of our own country not only advocate 

 the claims of these series to a place in analysis, but even attribute to them, after tlie 

 example of Euler, finite numerical values. 

 For instance, it is affirmed that 

 I 1—3 + 5— 7 + 9— 11+ &c. =0, 



; that 



1 1 — 1.2 + 1.2.3 — 1.2.3.4+ &c. =-4030 



I and still more strange that 



I 1 + 2+4 + 8 + 10 + 32+ &c. =— I. 



! In a paper about to be submitted to the Royal Irish Academy, and of which the 



present communication is a brief abstract, I have examined all the reasonings by 



1 which these singular conclusions seem to be established ; and I have, I think, shown 



i that such conclusions are in fact at variance with the analytical principles which 



have hitherto been appealed to in justification of them, viz. common algebraic deve- 



I lopment, the differential theorem, definite integrals, &c. The following are two of 



I the general principles established in the paper of which this is an abstract : — 



I 1. Whenever an infinite series becomes divergent for particular arithmetical values, 



I what has generally been called the generating function of the series requires a cor- 



! rection, which cannot be disregarded without committing an error infinite in amount. 



2. And that so far from such series being, as usually affirmed, always algebrai- 



' cally true, though sometimes arithmetically false — considered in reference to the 



i generating function — on the contrary, they are always algebraically false, though 



i sometimes arithmetically true ; true, namely, in those cases, and those only, for 



which the algebraic function omitted becomes evanescent. 



On a Principle in the Theory of Probabilities, By Professor Young. 



Let Pj, p^, ^3 ... pn be the respective probabilities of happening of n independent 

 i events ; then the following general principle will have place, viz. — 

 I l'i+F2+i'3+ • • • +i'n= the prob. of one of the events at least happening. 



+ the prob. of two at least happening in conjunction. 



+ the prob. of three at least. 



+ the prob. of all happening together. 



This general principle, Mr. Young observed, has not hitherto been noticed. It 

 affords an intelligible interpretation of the sum of the probabilities of any number of 

 independent events, and it is, moreover, useful in enabling us very readily to deter- 

 mine certain compound probabilities, when others are already known ; thus, let there 

 be but two events ; then by the above principle 



Pi+Pj:^= prob. of one at least happening, + prob. of both happening. But the pro- 

 bability of both happening is known to hepi, p^, 



•'• P\-\-Vi — Fi V"'^^ prob. of one at least happening. 



1844. " B 



