iga PHILOSOPHICAL TRANSACTIONS. [aNNO 1771, 



Or, putting n = — ~; a =/n, &c. Then l. 



- = 2s. 

 P ■' P 



Where p = 1 ; n = — — ; let a = S/n, &c. Then l. r = 



r + 



r — 1 



Or, in this case, putting n = —— = a; b = an^, &c. Then l. r = 2/s. 



Where j& = 1, anci/= 1 ; n = ;rT~; l^t a = 2n, &c. Then l. r = s. 



10. In 3 quantities p, q, r, increasing by equal differences, the logarithms of 

 any two of them being given, the logarithm of the 3d is also given. 



1. For l. ^ = 2/ X (v — x) = 2a — (p + k) 



= 2/ X (W + iN* + -J-n" + 4-N« &c.) = 2/y. Where n = J-=^. 

 Or L.?? = L. -^^ = 2/y = 2q - (p + r). 



pr qq — vv •" ^ ' 



Because l. -?^ =2/X — ^. 



2. Putting n = ^^-I^ z= — ^- — = (where i; = l) — ■ — ; a=/n; b = an^ &c. 



o qq + pr qq + rp ^ 9? + /"" 



Then l. - = 2s = 2a — (r + p) ; or a — — "^- = s. 

 For since w = qq — pr =■ 1 ; put 59 for r; pr for y. 

 Then r — p = qq — pr = w = 1; r -\- p = qq + pr. 



3. Putting n = - = A, &c. a = 





— |a — iZ> — ■J<;, 



c = i — ia — i^, 



e = -iV— T« — T* — ic — ^d, 

 &c. "^ 



And M = GA + iB + cc + dD &c.; 2 = ^ (r + p)* A = 4. (r — p). Then a 

 = 2 + AM. 



For G — p =/v = «; K — P = 2/z = 2A; but (^ = 1 + m = t~t;; there- 

 fore, &c. 



11. Any numbers/), q, r, &c. and as many ratios a, b, c, &c. composed of 

 them, the difference of whose terms is 1 ; as also the logarithms a, b, c, &c. of 

 those ratios, being given : to find the logarithms p, a, r, &c. of those numbers, 

 where the form is 1 . 

 For instance, if /> = 2, 9 = 3, r = 5, 



« = a=)|; ^ = (H=)£; c=(^=)^. 



Now, the logs. A, B, c, of these ratios, a, b, c, being found, the log. of either 

 1, 3, 5, or of any number compounded of them, may be found directly, by 

 making each successively equal to a", by, c". Thus, for the log. of 10 = 2.5. 



Let a^b>(^ = 1^ X aTJ ^ 3^F = ^'''•^"^'' ^ 2^.3-^5-> X 5".3-«.2-3«=2.5. 



