34 ^'''' Vince's Supplement on the 



jiator * ; and it may alfo be added, that the fame feries arlfes- 



from —— --- ,"^ . ' , provided the number of units be g-reater in 



I -f- I -|- I -J- I -j-OtC. '■ o 



the denominator than in the numerator. The ccrre6lion will 

 therefore be different in different circumftances, and will depend 

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

 the third part of my paper, I appHed the correction to thofe cafes 

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

 where the corre6lion is in general as I tliere gave it ; but as I 

 did not apply my method to any other feries, 1 confeis that it 

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

 ferent, which it neceffarily would had [ extended my reafon- 

 ing to other cafes. 1 fliall therefore add one example to (hew 

 the method of corredlion in other inftances, where the value 

 of the correction will be found to be different, according as we 

 beoin to collect at the firlf or fecond term-. Let the leiies be 

 .* _ 1 4. 5 _ 6 _[_ 8^ _ ^c. fine fine^ which camie originally from 



___J ^ ; DOW if we bepin to colled at the firfl: term, the feries 



becomes ~— H — ^ — V &c. and for the fame reafon as before, the- 

 1.2 4-5 



correaion, to be added, is ]. ; but — — + — - -1- &c. ~ \- of a 

 » 1.24.5 



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

 quired =±A+ 4-. ■ If we begin to colled at the fecond term the 



feries becomes 2 -" -^ &c. ; and the corredion to be 



2.4 5.7 



fubtracled is 4 ; for the fecond part of the original feries is now 

 _ I 4- I _ I ^_ I _&c. which was produced by - , -; but 



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

 OND, made the fame obfervation, 

 J 



