1889 - 90 .] Lord McLaren on Equation of n th Degree. 
273 
Xoy, rri, and known coefficients, by which Xoy and rri are first 
found, and afterwards x. 
Solution of the Equation of the 5th degree. 
It is known from the theory of equations that by a not difficult 
operation any quintic equation may be reduced to three terms. I 
suppose this preliminary reduction performed, and the equation 
given in the form 
ax 5 ± /&e 4 = 1. 
If we take the upper sign there are two possible solutions derived 
from the relations, log ( ax 5 + fix*) = j + 1 
- 1 =10 gPxt + Xoyif (b). 
If the equation has more than one real root, which is not generally 
the case where the two powers of x are both positive, one of these 
may be found from each form.* If there be only one root, the first 
formula will be used if the quantity ax 5 be < /3x\ and vice versa, as is 
usually evident from the values of the coefficients. 
(1) From the given equation we have log (ax 5 f fix 4 ) = 0, and 
from formula (a) we have, as above, 
log (ax 5 4- J3x 4 ) = 0 
- log x = 
log a f 5 log x + Xoy ! ; 
Xoy 4 -flog a 
5~ 
(i). 
The equation of the argument, mL of \oy\ is m! = log (ax b ) - log (/£c 4 ) 
- log a? = log a -log/?- rrt . . . . (2). 
By equating the values of logx in (1) and (2) we obtain 
5m! + Xoy! = 4 log a - 5 log /?, or 
iTli + i ^-oyl = 5 log a - log P — 1 .... (I.) 
This may be considered a formula of reduction for the 5th degree. 
The character of the solution then is : — The equation ax 5 + /fcc 4 = 1 
is made dependent on the solution of the transcendental equation 
V + 5 -4>y — c ) while the latter is soluble by reason that tables of cf>y 
(i.e., log ^1 + have been computed. 
The operator is now to look in the table of Logarithmic Differences 
* If the equation has two real roots, it must have three. 
VOL. XVII. 
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