3^ T ) r , Allman's Methods of clearing Equations of 
i. e. x^rn" — o^x^ymn + ^xy'^mn — y^n" + = x^m — y^n . cz ; 
or,x'w + 3^®7z . xm — ; 9 ^x'^m-\-y'^n .yn — : x^m — y^n — cz\ cz = o. 
Restore the values of x, jy, and z : divide ~c\ then, 
m' * 
14600 
27“' 
— ysabc. 
m 
4 — 
7850 
- ' ■■■■■ c 
27 
+ g^abc^ 
200 ^5 
~27 
^3 ^ ^2 
-f i^a^b^c. 
4- i\abc^. 
— 
m -{■za^h^c^ = 0: which is an equation of 7 dimensions, free 
-\-a^b^c 
from surds. 
In like manner, the extermination of 3 surds of the 1 ith 
power from an equation might seem to require the solution of 
an equation of 5 dimensions: but in this case, the highest 
term, if I may so speak, vanishes ; so that an equation, con- 
sisting of 3 surds of the 11th power, may be freed from surds, 
without the solution of any higher equation than a biquadratic. 
The labour however is great. 
As preparatory thereto, and not to refer elsewhere, the 
solution of a biquadratic equation may be here given. 
Suppose it, as any equation may be so transformed, to want 
the second term ; thus, 
qx^ -j- rjc + «^ = 0 : suppose also, x'" -f- + / = 0 ; 
^ = 0 : then. 
x4 ^ 
-if* 
X +Jg = 0: 
and. 
f^g = q 
—ef+eg = r 
fg — s 
] 
Therefore, 
/-{■g-e^ + q 
-f+g = — 
V 
/= 
O' 
£> 
e^j^qe—r 
2 e 
e'^-{-qe-\-r 
ze 
fg 
e^-\-zqe^-\-q'^e ^ — r 
then, + '2qa* 
_ 4 S. 
0. Put e'^^ ~ z ^ ^ ; then. 
