the Sums of Neutral Series. S65 



choice of these will obviously depend the sign with which 



r-t-^^-^-Viv^ or, which is the same thinff, --^ is to be taken: — it 



will be minus for the first mentioned value of [2.], and plus 

 for the other, so that the sum of the proposed series is 



2 \3 2/ ~ 12* ' ' l\^ 



The term neutral series, a term I believe first adopted by 

 Hutton, should I think be in strictness confined to the forms 

 which arise, as in this example, independently of all connec- 

 tion with, and therefore uncontrolled by, the laws which goverii 

 general algebraic series. .jjuului eJi iafjj>3o iq lijul//, 



It may not be superfluous to remark, iri pli^sing, that 'the 

 Series here discussed is very intimately connected with the 

 doctrine of definite integrals, a doctrine into which a good 

 deal of error will be found to have crept. A very able con- 

 tributor to this Journal, Mr. R. Moon, Fellow of Queen's 

 College, Cambridge, has, I see, recently turned his atten- 

 tion to this important topic (Phil. Mag. vol. xxvi. p. 483). It 

 will be seen however, from the present communication, that I 

 have been constrained to differ from him, as to the general 

 theory of the series 1 — 1 +1 — 1-4- &c., regarding that series 



as strictly and exclusively — , when it is the limit of the con- 

 verging cases of I -x- + x'^—a:^+ &c. I will merely add fur- 

 ther, that I invited the attention of analysts to this subject, in 

 connexion with non-converging series generally, in a paper 

 read before the British Association at the York meeting in 

 184-4, a short abstract of which appears in the volume of 

 published reports: the same paper was also read at the 

 Royal Irish Academy in January 1845, and is published at 

 length in the Proceedings. 



It is perhaps scarcely necessary to state, that although in the 

 preceding reasoning I have considered 1—1 + 1—1+ &c., 

 when really a limit, only in connexion with l — x + x^^—a^-t 

 &c., yet the same reasoning applies when it is viewed in con- 

 nexion with 1 —x'^ + x^—a:^ + &c., or in general with 1 —x* 

 + ^-^ — x'y + &c.; and, when the series is strictly neutral and 

 isolated, that I regard the assumption, which would connect 

 it with any particular case of this last general algebraic form, 

 to be no more unwarrantable than the assumption ordinarily 

 made of its connexion with 1 — a; + x'^~x^ + &c. It is this 

 assumption which will be found to vitiate several important 

 and generally received results of analysis, for it is only when 

 Jj^ie assumed connexion .act.iwJl^ ^xjsts, that the unique value, 



