34 -Mr*. Vince’s Supplement on tie 
nator * ; and it may aifo be added, that the fame femes a rife s 
£ rom t ~ '^ ^ 7 ? provided the number of units be greater in 
the denominator than in the numerator. The correction 1 wilt 
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 applied the correction to thole cafes 
where the original feries arofe from the expanlion of a binomial, 
where the correction is in general as I there gave it ; but as I 
did not apply my method to any other feries, X 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 fhall therefore add one example to Ihew 
the method of correction in other inftances, where the value, 
of the correction will be found to be different, according as we 
begin to colleCt at the firft or fecond term. Let the feries be 
— 4. 4-p — f -f \ — &c. fine fine, which came originally from 
1 now if we begin to colleCt at the firft term, the feries 
becomes — — + — — + & c. and for the fame reafon as before, the 
1.2 4.5 
correction, to be added, is. 4; but — — + -- + &c. = 4 of a 
1.2 4*5 
o t - 
circular arc (A) of 30? to the radius — ^ ; hence the fum re- 
quired = * A + 4. If we begin to colleCt at the fecond term the 
feries becomes 2-——-— &c. ; and the correction to be 
2.-4 5.7 
fubtraCted is 4 j for the fecond. part of the original feries is now 
„ ! 4 ! _ ! 4 1 _ &c. which was produced by ; but 
* X have been fince informed by Mr. Wales, F. R. S. that a. pupil of his, Mr* 
"jjqkd, made the fame obfervation. 
a 
2 
