q 6 Dr. Allman's Methods of clearing Equations of 
which put == — . w: multiply by h: then, b^\/ab' 
= — bm: this quadratic equation gives, ^\/ab* = — ^ b ± ^ 
ah' = d!l±^ + ^ ^ah + 6 * 
— of in =: ± b — m V b* — ^bm : which squared gives, /^a^b^ 
j^ab^ — i2ab*m — 6 b^m -j- Qb^Jn^ = E — Gb^ih -|- gb‘m* 
— •.* a% + ah' ~ ^abm + == o ; i. e. nd — abc = q ; 
\* a -{■ b c =£ Q'yabc. 
But an equation consisting of 3 cubic surds only, msy be 
cleared of them without the solution of a quadratic equation,, 
thus : 
Let + ^Vb + *\/c = 0 : then, (2;) ^ a'b 
ah' ■=. — c, i. e. since ^^a + ^\/b = — ^ 
^x/^bc = — r: therefore, (3;) a + b c — ^^s/abc; ahd, 
« + 6 -|- r = Qyabc. 
And, universally, an equation consisting of 3 surds only, 
whose common index is any o’dd number, may be cleared of 
them, if we admit the solution of an equation, whose highest 
dimension is half the index of the surd minus unity. 
Because in any integral power of a binomial, as the co-effi- 
cients of terms at equal distances from the extremes are equal, 
those ferni's rhay coalesce, the compound factor being equiva- 
lent to a sirh’ple one, as may be more fully seen below. 
To return to cubic surds : s d'e v*\/ de' = o, then, 
by the last example, (3 ;) s^ fd'e -j- '^^de' — ^stvde. 
Lk a ^ ^ • then, (3;.) a 6 -f 
= 3 ; 
and, ^z4-6 -{-^+^ + gWd^e 3 *\/de'^ = 2yab . 
c r> \ 
d +e + S ‘</d'e + 3 ‘ \/ de* I put a — |— b “1~ d + e = gr; then. 
