278 
Proceedings of Royal Society of Edinburgh. 
[» 
II. Where in the given equation the terms of x have contrary 
signs , or 
ax n — fix 71 ~ p =l . 
Case (3). Log {oaf - /3x n ~ p } = 0 = log a + n log x - XoyZ _ p ; 
Also, 
whence. 
loga? = 
loga? = 
_A^y^-loga 
m^-p + log _£-loga 
p ’ 
/p\ x (n - p \ _ . f Formula of 
VCln-p ~ J^oyn-p = ^ n J l°g a ~ P = I ^ Eeduction 
For the indices n and (n - 1), this becomes 
(1) . 
( 2 ) . 
(C). 
m:-i-(^)Aoy^_ 1 = (^-) 1 °ga- lo g/3 = g . . (Cj). 
The formula (C) is to be used with signs changed when necessary 
to make q positive. 
The quantities rq and Xoy are found from the table of Loga- 
rithms of Subtraction. These observations apply also to formula 
(D). 
Case (4). If fix n ~ p be the greater of the two terms , we find 
/ 1 \ { n „ i ( Formula of ) . 
WT' = J log /» - log « - 2- { Reduction / P)- 
For the indices n- 1 and n, this becomes 
mT 1 + (^l) Xo ^ r 1 = (fh ) l0g Z 3 - lo 8 “ = 2 ' ( D i)- 
It is evident that the solutions given for the quintic equation are 
only particular cases of the three-term equation of the nth. degree. 
Example — 
x 7 + 3x i = l . [» = 7; = — =i 
a = 1 ; p = 3. 
As x is evidently fractional, the term 3a? 4 is greater than a? 7 , and 
we must use the formula of reduction (B), or 
flog 3 -log (1) = vni - f Xoyi 
I. Log a = 0 ; log /? = log 3 = 04771213 
<? = f log /3 = 08349623 = rav- f Aoy 4 . . (a). 
