34 ^^* Vince's Supplement on the 



iiator * ; ^nd it may alfo be added, that the fame feries arlfo'^ 



from — — -~ — - — r-T— > provided the number of units be greater iii 

 the denominator than in the numerator. The ccrrecSlion will 

 therefore be different in different circumflances, and will depend 

 on the nature of the quantity which was at firft expanded. la 

 the third part of my paper, I applied the corredion to thofe cafes 

 where the original feries arofe from the expanlion of a binomial, 

 where the correftion is in general as I there gave it ; but as I 

 did not apply my method to any other feries, I confefs that it 

 did not appear to. me, that the correction would then be dif- 

 ferent, which it neceflarily would had I extended my reafon- 

 ing to other cafes. I (hall therefore add one example to (hew 

 the method of correction in other inftances, where the value 

 of the corredion will be found to be different, according as w^e 

 begin to colleCl at the firft or fecond term. Let the feries be 

 _*__l + l-^-}-l- &c. fine finey which came originally from 



\ ^ : now if v/e berin to colle6l at the firfl term, the feries 



becomes — — -1 — — +&c. and for the fame reafon as before, the 



1.2 4.5 



correaion, to be added, is 4 ; but — — + — - + &c. = i of a 



1 • - 4 • 5 



circular arc (A) of 30° to the radius — ^ ; hence the fum re- 



quired = 4-A + -i. If we begin to colle£l: at the fecond term the 



feries becomes 2 -" ? &c. ; and the correftion to be 



2.4 5-7 



fubtracled is -| ; for the fecond part of the original feries is now 

 _ I _i- 1 _ I 4. 1 _ &c. which was produced by - ]_ 2 , ; hut 



* I have been fince informed by Mr. Wales, F. R. S. that a pupil of his, Mr. 

 »;OND, made the fame obfervation* 



3 ^— 



