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 LIX. Intelligence and Miscellaneous Articles. 



REMARKS ON SOME NEW ALGEBRAIC SYMBOLS IN PHIL. TRANS. 

 FOR 1862, PART I.* BY S. M. DRACH, F.R.A.S. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 

 T N Mr. B. Gompertz's paper " On the Science connected with 

 ■f- Human Mortality," the learned and venerable author proposes, 

 in p. 5 19, the new foot index of crosier shape as indicating a common 

 or hyperbolic logarithm, or their antilogarithms, instead of the usual 

 exponential bases e and e. The four positions of this mark «— having 

 each a separate meaning, which might be mistaken by an error of 

 the writer or printer, and as its attachment to a binomial or multino- 

 mial would be attended with some indistinctness, I beg to propose the 

 following logarithmic mark, viz. J before the quantity of which the 

 logarithm is wanted, and "| before the quantity (a logarithmic one) 

 of which the natural number is required. Whether the Briggean 

 or Napierian log is wanted may be indicated by a prefixed b or n ; 

 and the following synopsis will give an idea of the powers of my 

 notation. 



b*J a = (com. log. of a) raised to the ith power. 



b J a*== com. log. of (a i ) = i (bj a). 



b'~| a = ith. power of number whose common log is a. 



b "7 a l =: number whose common log is a 1 . 



n*J a = (nat. log of a) raised to the ith. power. 



nj a}\ n*""| a; n"| a 1 ; similar to b, changing "common log" 

 to " natural log." 



Hence we may combine n fe J b*J a = kth power of nat. log of a 

 number N, which N=ith power of com. log of a natural number a. 

 We might further apply J to sines, &c, sin* J d=ith power of sin0, 

 cos* J sin^^&th power of cosine of an arc = a number N, which N 

 is (sinfl)*; similarly tan™ J sm i d=mth power of tangent of an arc 

 = N ; sin J 0*=n*J a= sin of (arc dy= kth power of nat. log of a 

 nat. number a, &c. 



Thus to the recognized > < J_ II > the symbols ~| J might fitly 

 be added. Mr. Gompertz's embodiment of zeros is quite a boon. He 

 puts -00000000763 =(¥)763, and 89600000=896(5); theformer 



shows that 7 is preceded by eight decimal zeros, the latter that 6 is 

 followed by five zeros before the decimal point. This suggestion is 

 particularly valuable in large and approximate results, as the number 

 of vibrations of a light-ray in an inch, the English mile radius vector 

 of a planet from the sun, the English mile distance of a fixed star, 

 parallax = 1", which is 206000 times 95 millions of miles = 1957(H)) 

 miles in Mr. Gompertz's notation. There are two errata (p. 517, 

 line 26, for L m+n read L m+2n ', and p. 543-4, for x° read S# ) that 

 may be pointed out. 



* Erratum in page 367, line 14, for 4° read 44° (Mr. De la Rue's paper). 



