Calculation of Logarithms by means of Algebraic Fractions. 479 



seen during the late summer, to describe appearances with 

 accuracy, and refer them to their real causes, independently 

 of any theory. Believing that glaciers once existed in Scot- 

 land and in the north of England, I thought it probable traces 

 of them might be found in Wales also ; and I have shown 

 that not a few appearances there may, on a hasty survey, be 

 referred to such an origin. But believing, after deliberate 

 examination, that these have either been produced by other 

 adequate causes, or could not have been due to glacier action, 

 I have felt myself bound honestly to state the conclusions I 

 have arrived at, being satisfied from some experience, that to 

 allow the observation or the judgement to be warped by pre- 

 conceived theory, however plausible, or to decide on partial 

 insufficient evidence, must be ultimately injurious to the cause 

 of truth. 

 Manchester, Oct. 18, 1841. 



LXXI. Calculation of Logarithms by means of Algebraic 

 Fractions. By the Rev. R. Murphy, M.A.* 



r T , HE logarithms here used are Napierian, which may be 

 •*- readily converted into logarithms of any other system, 

 simply multiplying them by the modulus. 



Take the coefficients of a binomial raised to any positive 

 and integer power w, and the same number of the coefficients 

 of a binomial raised to the next superior negative power, 

 — -{n + 1) ; multiply both sets, term by term, multiply the pro- 

 ducts by the successive powers of a number as 1, t f t% ...t n , 

 and connecting all the terms thus formed by the sign -f , this 

 sum is the denominator of the fraction. 



2 

 Multiply the last term but one of this denominator by — , 



2 2 2 



the last but two by — -J r, the last but three by — 



w n n— 1 J n 



2 2 . 



-\ - + -, and so on ; connect the products with the 



sign + , the sum thus obtained is the numerator of the frac- 

 tion. 



The fraction thus formed is to be added to log t in order 

 to obtain log {t+1). 



The sum is always deficient. 



Call the above denominator P n and put T = 4 / (£+ 1), the 



• Communicated by the Author. 



