$2 Dr, Allman's Methods of clearing Equations of 
multiplied by 6^, gives + 26 + 26" W a*h' + 6* 
V ah^ = — Wm, a biquadratic equation of the quadratic form, 
thus, W a'b* bW at/ b * . Wa^b* + bWab^ = — which 
solved, gives Wa^b* + b ^Vab* == — i ~ , another 
quadratic : then = — ■“ ± 2\^b*—^bm . 
to the 5th power, and the equation will be cleared of the radi- 
cal sign of the surd of the 5th power : thus, 
ab*= — 1-6*-}-“^ b*m^^ EV b* — /^b^m ^ ^ b^m^ — 6^ x 
divide by -L 6^, and transpose ; then, 
6* — ^bm 2ab z= V b* — ^bm if m ^ — 6“ + 26 V b"^ — 4pm, 
Square, transpose, and divide by 26 ; then, 
^3 ^ 5 "^- ^2 ^loam. b == -± b' — gbm V b^ — ^bm. 
+ 2a*. 
Square again, transpose, divide by 46, and arrange: 
then, ot’ * — Qsabm^ +%“ab‘. tn = o. 
+ab* 
But, as in the case of 3 cubic surds, a simple equation sup- 
plied the place of a quadratic, so, when an equation consists of 
* 
3 surds of the 5th power, a quadratic may supply the place of 
a biquadratic, or of two quadratic equations. 
Thus, + Wb + V<^,= 0, *.• (2 ;) ^ + ^ + a*b lo 
^\/a^b^ -h 10 Wa^h^ -J- 5 = — r, (^^\)Wa*'b -[- 2 Wa^b* 
+ a^b'^ + W ab^ =■ = — mi 
i>i/ah^zz^>^ah . ^ r= s^/ab . — ^ \/ ahc^ 
But 
{ 
n — ^^a^b^ . 5y'a-j-5y'6 ~ —^^a^b* . — ^v/c =: -{-^\/a^b^c 
Therefore s^a^b'^c — ^\/abc^ :zz m. 
This, multiplied by r, gives the quadratic equation, Wa^b'^c^ 
] 
