VOL. LXXIX.] PHILOSOPHICAI. TANSACTIOIfS. 567 



rx(r-yVx(r-4y-)x(r-y'>x&c . ^ , ^^^_.^ j^^_ ^^^^^^ 



(;. _ pr) _ (r - 3p') X (r - 5/>'") X &c. . ' y ' 



co-efficients of the alternate terms of the binomial theorem, viz. s = n , 



!L__i, / = 7z.— — •— 3— • — — , &c., and i = w, i =n.— — . — - &c.the 

 co-efficients of the remaining alternate terms ; the numerator r X {r — 2p') X 

 (r — 4py X (r — 6py" X &c. = (if n = 2"-') r" ~ pj&r^-' + q/>V«-» — 

 j{p3^N-3 ^ , . L/j"-! X r''-''+' + uip^r^-" + &c. and the denominator (r — /))' X 

 (r — 3pY X (r — 5py" X &c. = r** — pj&r^-' + apV^-^ — KpV''-3 -f- . . . 

 j»,n-i^N-»+i (-j-M± 1.2.3..W — 1)/)^"-" ip &c. whence the numerator and 

 denominator have the n first terms the same, and the next succeeding terms differ 

 by 1 . 2 . 3 . . (n — l)p"r^-" ; the numerator divided by the denominator = + 

 Ll1i1^—S12 p» nearly, if r be a great number in proportion to p, &c. it would 

 be + when n is an odd number, and — when even. 



8. The logarithm of the fraction k by the common series = k — 1 — 

 ll:ii2' + ^^^- — &c. has for its first term =: + ^' ^' ^^ . r" "" ^ X j^'nearly; 

 for its 2d term the square of the first divided by 2, &c. 



9. The error of this equation not only depends on the logarithm of k, which 

 may be calculated to any degree of exactness, but in the calculus on the errors 

 of the given logarithms. 



10. If r be increased or diminished by any given number, the n first terms of 

 the numerator and denominator will still result the same, and the next succeed- 

 ing terms will differ by 1 . 2 . 3 . 4 . . n — 1 X /)" X r** -«. 



11. Let n , — -— numbers be 2, n , -—- . — r-- . — 7- numbers be 4, w . 



2-11-1 . " "" . " "" - . - 7 . "-^- numbers be 6, &c. ; their sum, the sum of 



2 3 4 5 ' ' 



the products of every 2, the contents of every 3, 4, 5, &c. to n — 1 of them, 

 will be equal to the sum, the sum of the products of every 2, of the contents 

 of every 3, 4, 5, &c. to w — 1 of the following numbers, viz. n numbers 



,,, »— in — 2 1 1. 1 o »— In— 2»— 3re— 4 



which are 1, n . — j- . — j- numbers which are 3, n . — - . —— . — — . ~- 

 which are 5, &c. ; and the sum of the contents of every n of the former will 

 be less than the sum of the contents of every n latter numbers by 1 . 2 . 3 . 4 

 , .n — 1. 



12. The method given in art. 4, which I name a method of correspondent 

 values, easily deduces and demonstrates the preceding equations, which cannot, 

 without much difficulty, be done by the preceding method of differences ; the 

 method of correspondent values is much preferable to the method of differences, 

 both for the facility of its deduction, and the generality of its resolution : for 

 instance, from this method very easily can be deduced, &c. the subsequent and 

 other similar equations. 



