312 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



Elizabeth; but such as it is, I have thought proper to examine it, and find as 

 follows : 



Inches. 

 On this rod, -£., or the 1st foot, is equal to 11.973 on Troughton's. 



the 2d foot is 11 .948 



the 3d foot is 12.047 



The mean foot is 1 1.989 



\ yard, or 18 inches = 17-S46 



£ yard, or 24 inches = 23.92 1 



I yard, or 27 inches = 26.937 



■J yard, or 31 J inches = 31.443 



-f-§- yard, or oof inches = 33.665 



Entire yard, or 36 inches = 35-966 



And the mean yard as 35.924 



Mean — .076 

 And by so much Mr. Troughton's measure is the longer. 



And the probable error, in the divisions of this old standard, is about -j-f-^ inch. 



It may now be desirable to see the comparative lengths of these various stan- 

 dards and scales, reduced to one and the same measure, viz. Mr. Troughton's. 



Inches on Probable error 



TroughtonV Difference. in divisions. 



36 inches, on a mean, of Hen. 7th's standard of 1490, are equal to . . . 35.924 — .076 .03 



of standard yard of Eliz. of 1588 36.015 4- .015 .04 



of standard ell of ditto, of 1 588 36.016 -f .016 .04 



-* of yard- bed of Guildhall, about 1660 36.032 4- .032 



*of ell-bed of ditto, about 1660 36.014 + .014 



*of standard of clock-makers' company, 1671 35.972 — .028 



— — — *of the Tower standard, by Mr. Rowley, about 1720 36.004 4- .004 



' of Graham's standard, by Sisson, of 1742, viz. line e = 36.0013 4- .0013 



of ditto, ditto, viz. line exch = 35.9933 — .0067 



i of Gen. Roy's (Bird's) scale c all made, probably, "^ = 36.00036 4- .00036 .0003 



. — of Mr. Aubert's ditto, ditto I between the years > = 35.99880 — .00120 .0006 



of Royal Society's ditto, ditto I 1745 and 1760. J = 35.99955 — .00045 .0004 



■ of Mr. Bird's parliamentary standard, of 1758 = 36.00023 4- .00023 



■ of Mr. Troughton's scale, in 1796 = 36.00000 .00000 .0001 



Hence it appears, that the mean length of the standard yard, taken from the first 

 7 instances in this table, agrees with the quantity assumed by Mr. Bird, or Mr. 

 Troughton, to within -j-oVo inch, but that the latter is the longest. 



IX. A new Method of computing the Value of a Slowly Converging Series, of 

 which all the Terms are Affirmative. By the Rev. John Hellins, F. R. S. p. 183. 



1. The computing of the value of the series ax -\- bx 1 -j- ex 3 -f- dx 4 -f- &c. ad 

 infinitum, in which all the terms are affirmative, and the differences of the co- 

 efficients o, /', c, &c. are but small, though decreasing, quantities, and x is but 

 little less than 1 , is a laborious operation, and has engaged the attention of some 

 eminent mathematicians, both at home and abroad, whose ingenious devices on 

 the occasion entitle them to esteem. Of the several methods of obtaining the 

 value of this series, which have occurred to me, the easiest is that which I am now 

 to describe, by which the business is reduced to the summation of 2, or 3, or more 



* These four quantities are taken from Mr. Graham's account, in the Phil. Trans, vol. 42. — Orig. 



