Dr. Waring on the 
1 7 * 
R/r*-r+ . . . L/ M - * I r N '~«+ I ( = i=:M+ i .2.3.. 0- 1) ^y N -*=p 
&c., whence the numerator and denominator have the 0 firft 
terms the fame, and the next fucceeding terms differ by 
1 . 2 . 3 . . n — 1 . the numerator divided by the denomi- 
nator = 1 ± 1 , • 2 * 3 j ~ _ I pn near }y 5 if r be a great number in 
proportion to p 9 &c. it would be + when 0 is an odd number, 
and — when even. 
8. The logarithm of the fraction K by the common feries 
= K — 1 — — — - &c. has for its firft term — =±= 
2 3 
nearly; for its fecond term the fquare of 
r n 
the firft divided by 2, &c. 
9. The error of this equation not only depends on the loga- 
rithm of K, which may be calculated to any degree of exact - 
nefs, bu£ in the calculus on the errors of the given loga- 
rithms. 
10. If r be increafed or diminifhed by any given number, 
the n firft terms of the numerator and denominator will ftill 
refult the fame, and the next fucceeding terms will differ by 
i . 2 . 3 . 4 . . 0 - 1 x p n x r N '“'*. 
T . n — 1 t 1 A it — 1 n— 2 n—7 
11. Let n . numbers be 2, n . . . — - num- 
2 234 
bers be 4, n . . Lz? . n JZ 3 . n -ZA # numbers be 6, &c.; 
their fum, the fum of the produdts of every two, the con- 
tents of every three, four, five, &c. to n — 1 of them will be 
equal to the fum, the fum of the products of every two, of 
the contents of every three, four, five, &c. to 0 — 1 of the 
following numbers, viz, n numbers which are 1, n . n — . hzi: 
2 3 
numbers 
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