quadratic i cubic , quadrato- cubic y and higher Surds, 27 
r + 's/d*e W =■ ab , d e ^ s/d^^ + 
+ 3 ^^ 
i.e. +yf- Wd'e -i-frfr! "\/de" = ab.d-\-e->fS‘ib'-\/d'^+S‘>^Vde‘; 
+ de' 
+ 3 rf<!. 
-^ide. 
+3^*'o +3'"*'o 
.put — ^z6 = X, and d-\-e"=y\ *.• (4;) -\- 6 der -\-^er.^/d^e -^zdr.\/ de'zrzo. 
+ xy + 3 X . ' + 3 f • 
Substitute now, for 5, -|- 6 der - 1 - a:y; for t, -f- 3 ^^ + 3 -^; 
for Vy ^d ^^dr + 3a: ; and the equation, dd*e -{- v^dt* — 
^stvde =z 0, will become, 
yzjd'^c^y*- , 
; — 9^^/ 4- 3^jyr r ^ 63 t/«rx>-. ^ —zjdex^, ^ ’^^laexyr 
— ^dex^y*r -f- =.0y 
which is an equation of 9 dimensions, z. e. of 27 times Jhe 
height of the given one ^y/a + ^\/^ + ^Vd + 
If another independent cubic surd were added, the equation 
resulting free from surds would be of 27 dimensions, or of 81 
times the height of the given equation. 
Universally, if an equation consist of any number, of inde- 
.pendent surds having a corpmon index, the equation resulting 
free from surds will be so many times the height of the given 
one, as there are units in the common index of the surds raised 
to the power whose index is the number of independent surds 
diminished by unity. 
As the solution of a cubic equation is required for the ex- 
♦ 
termination of some of the higher surds, it may be worth 
while to shew the connection of the rule, called Cardan's, 
with the extermination above given of cubic surds. 
If + ^s/c = Oy then a b + c = ^^^abc, as 
before; or, by substituting x,y, s;, for y/b, \/c, respec- 
tively , if a: - 1 - y + ^ = 0, then -f. ^ = 3-ty^. Suppose 
E 2 
