C 46X ] 
II, Any numbers A $'> many ratios 
b, c, &c. compoled of them, the difference of whofe 
terms is i ; as alio the logarithms A, B, C, &c. of thofe 
ratios, being given : To find the logarithms P, Q, R, &c, 
of thofe numbers, where the form is i. 
For inflance, if p~2, §'=3, ^=5, 
Now, the logs A, B, C, of thefe ratios, c, being found, 
the log. of either 2, 3, 5, or of any number compounded 
of them, may be found diredlly, by making each fuccef- 
fively equal to a^yby^c’^. 
Thus, for the log of io=:2.5. 
2 .V 2 « 
Let ab^c^ = 1 x X -i- 
3 y y s5 s5 
2 3-5 3 -2 
3* ''.2~3^ X X 5* *.3—*'. 23 *1=2.5, 
Therefore 3^-3* — i x 3^^~->'— *x i. 
Confequently aj/^^x — 3s — 1=0; 2x — -y — z:^o; 2z~y-iro 
Therefore at = lo 3 y— 135 z — j i 
and a^° x X ff = (2 x 5 =) lo* 
Therefore 10 A -|- 13 B 7C=;log. of 10, to the form r. 
Or, fince — Tj b z=z — ^ c — 
2^ 3.5 3.2^ 
Therefore A-r2Q—3Pj B— 4P— Q^R; C—2R-Q-3P. 
Confequently P=:3 A-f-qB-f-^Cr^log. of 2 "I 
Q= cA-}-6B-|- 3C=:log. of 2 ito the form i. 
R=7A+9B+5C^log. of 5J 
Therefore P-j-R— loA-pi^B-j-yC— =log. of (2 X 5—) to. 
And J^P, y^Q, yR) are the logarithms of 2^ 3, 3, relpec- 
tively, in the fcale of logarithms whofe form is /. 
VoL. LXI. N n n 3 XLVIII. An 
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