274 
Proceedings of Boyal Society of Edinburgh. [sess. 1 
for values of rri and Xoy which satisfy the relation (I.). The nearest 
tabular places may be taken out at once by a mental computation 
identical with that which we perform in long division * in finding 
the successive figures of the quotient. This slight mental effort 
might, however, he saved by preparing a four-place table, in which 
the value of rrt would be given to the argument q for the 3rd, 4th, 
and 5th degrees respectively. The fifth figure would then be inter- 
polated ; as to which see note at the end of this paper. Xoy would 
then be found by the existing tables from rri as argument. The 
proportional parts are to be subtracted, because Xoy diminishes as 
rri increases. 
The quantities Xoy! and rnj being thus determined, x is imme- 
diately found from (1) or (2); but the possible error in the last 
decimal place, due to the imperfections of the tables, is avoided by 
using the form (1), in which the sum of the quantities has to be 
divided by 5 for an equation of the 5th degree, in order to obtain 
log x, which is the required solution. 
(2) From the given equation, ax 5 + /3x 4 = l, we also have 
from ( b ) 
log {ax 5 + pxt) = 0 = log (/3x 4 ) + Xoy 4 , whence 
-i og ,^ a). 
- log x = rri 4 + log a - log/? .... (2). 
5 log P - 4 log a = 4 trig - Xoy 4 ; ) Formula of 
iris - iXoyl = f log p - log a = q. J Reduction II. 
If the result of substituting the numerical values of log p and 
log a in the formula of reduction II. is to make q negative, the 
formula is to be used with signs changed. 
(3) For the equation ax 5 - /?x 4 = 1, we have 
Also, 
whence 
log { ax 5 - px*} = 0 = log (ax 5 ) - Xoyl ; 
. Xoy! - log a 
log X — ' 6 ■ • 
log x = TriJ + log P — log a ; . . . . 
5tri! _ ^°yl = 4 log a - 5 log p ; ) Formula of 
ml - i Aoyt = ‘ log a - log 13 = q. J Reduction HI. 
(!)• 
( 2 )- 
* In long division, when we mentally find a figure of the quotient, we solve 
an equation of the form y=ax + z. 
