54 > Allman's Methods of clearing Equations of 
the cube, being equal to half the index of the 7th power 
diminished by unity. 
Let Wa + Wc = 0 : then, 
(z;) « + 6-J-7 z \ + 7 — — s 
(3;} 5 T <^ah^— — — — - — - (put) =: — m 
But 
'^\/a^b + s^\r^^b^-i- ^A/ab . "^A/a + T^b =r — >yahc^ 
— zT \Ta^b'^—6’’ t/a^b^ (Py/a^b^ — z'^A/a'^b^ ~ — z"^ Aya'^b^J A/a-^-"^ A/b^—-\-zTA/a^b'^c^ 
+7^^3^4 _ ^ . Wa-VWb zz — 7ya^b^c 
Therefore — “^Aya^b^c -{- z'^ A/a^b'^c'^ — ’’Ayabc^zz — 
Multiply by — then, — sr c"" ’’\/abc^ ^ 
c~?n. Extract the square root of this cubic equation ; then, 
aWd^ * — c'*Vabc' = + cVm: this cubic equation, which 
wants the second term, when solved, gives obc^ = . 
+\cVm: m — -±^c\/m: — — ~c: 
27 27 
this, involved to the 14th power, will be free from all surds, 
except quadratic and cubic. 
Put m — + i \/m : -f i \Z;z == 5 , and + V^ri : 
— \a/}i z=z t‘. then, on dividing the equation by 
zzz ^\/ s 1 \ involve, then 3^ = ^x/s'"^ +14 ^v^s'^t + oi 
+ 364 1001 2002 + 3003 
+ 3432 ^\/ 3003 sY^ 2002^1/ + 1001 V 364 
+ 91 ^\/s’"V^ +14 Wsf^ + 
But, st = ^ m — ^ n z=z Ec -y ^^\/c=z:Wst,2iX\d^^Vc^=Ws^t''i 
then, by multiplication, ^ abc = 5* ^x/sf" + 145*^ + ^y's^t 
+ 3<34sV + 10016"^/“ + cioo^sY s"t + Wst^ 
+ 30035^/^ + 2002 S ^ t ^ sf + 10015V + 
364,9/'^ W s^t + Wsf + i4>^i* + Ws"t 
