VOL. LXI.] PHILOSOPHICAL TRANSACTIONS. IQl 



Then a will be found to be 2.30'2385092994 &c. 



And /= 0.43429448 190325 &c. 



If a = 1 =/, the form will be that of Napier's logarithms. 



6. Let B, b', be the logs, of the numbers x, x, in the form/= ^ , 



And N, n', the logs of the same numbers, in the form ^ = -. 



Then b^ = n/; bo = sx; bn' = nb'; b'^ = n^". 



For B : N :: (/ X - : «) X -::)/:«:: B : n :: - : - :: b' : n'. 



If X rr 10; B = 1 ; a— 2.30258 &C.; or/= 0.43429 &C.; « = ,p = 1. 



Then n = B X - = 2.30258 &c. 



« 



n' = b' X - = 2.30258 &c. X B . 



B 



b' = ^X n' = 0.43429 &c. X n'. 



7. Putting x=:9 + v;n = -. 



Then z = log. of x, or the log. of y + f, will be 



= + !L_Il + lL_il + ll_ll&cX/'- 

 — q 2g'- 3?' V — 5g' 6q^°^^' ^ J' 



= ± N — iN' + i-N' - ^n' + 4-N' — ^n'&C. X/. 



Fori = £,.*=£. (9 ± ,.) =/X ^=/X ^#^, 



= (^+ '"--% + "^ -^ + % -% &c.) X /. 



8. In 3 quantities p, q, r, increasing by equal differences, the logarithm of 

 any one of them being given, the logarithms of the other 2 are also given. 

 For, \eX V =^ q — p = r — q; n=- = l-HE = LZl-^ P,a,Bj thelogs.of/), q,r. 



l.L=i = (l.^-X-=)q-P=/X (N + iN' + iN^ + 4-N* + iN*&C.) = 



/v. ForL. -i-=/x -^. 



2. L.: = (L.^-=)R-a=/X(N-iN^ + iN^-iN* + iN*&C.)=/x. 



? 

 = /• X -— 



q -^ ^ q+V 



q ^ g 



ForL ^ + '' 



3. l: ^ =/ X (v + x)= B - p= 2/X (n + 4-N^ + iw' + -n' + iN» &c.) =1 

 2/z. Where n = (- =) ^. 



Ori..:=L.?^" = R-P = 2/z. ForL. i-ti:=2/X-^^. 



p q—v •' q — V •' qq — vv 



9. Hence, in two quantities, r the greater, p the less. 

 Putting n = ^—^■■, A = 2/n; b = an'; c = bn^; d = on", gcc. 



And s = A + -J-b + -f c + ^D + &c. Then l. - s= s; or r — p = s. 



