24 D/’. Allman’s Methods of clearing Equations of 
II. Let s/ d + Vh -“I- s/c -{- V d = 0 ; then 
(2;) a c ah ac + if he — d 
(3;) ^ + 6 + 6' — d^\- ei\/ ah = — ac — 9,1/ be \ squared 
gives, a -j- b -L c — d -|~ /^ah -|- ^ , a h -J- c — d , ah /^c . 
a b -j- 9>f ah 
(4O ei b — c — d' /yih — ^cd = 4. c d — a — b . ^ ab ; 
squared, results free from surds. 
NotCy universally: the last two surd factors vanish together. 
Surds, whose indices are integral powers of 2, may be treated 
as quadratic surds ; and the number of them, which may be 
exterminated from any equation, is equally unlimited. 
Let ^\/ a + *^b -j- ^'s/c = 0 : then, (2;) V ^ f- s/b + 2 
*'^ab — \/ c 
(3O + v/^ — v/^* = — 9‘'s/ah ; and, a-\-b-\-c 21/ ab 
— ac — 9^bc — 4-v/ ab. 
The surds, now all quadratic, may be thus exterminated: 
(4 0 ^ ^ “f” ^ ““ 2 ab 2 ac -p 2 y / be j u -{— b -|~ c -L ^a jy 
•1— 4 . ^ 4- r . i/ab = dgxc 4^r -j- ^c^/ab. 
■ " ■ ■! ■■'2 ■ ■ IM ■ III 
(5;) a ~\~b — c + ^ab =: 4 . ^ + b . ^ab : put a -\-h — c 
— 911 ; then -j- = 2/z + 4rV ' ab ; and n"" + 2^5/z“ + cC'b'^ 
“ 4/z* + i6cn 4“ ibV , ab ’ r n* — ab =. i6cn 4- ibV . ab 
; 
t\ e. — ^abe . 911 9C =z Sabc . a b c 
I - — - ■ - - ^ II. II, 
•.* a* 4 " ^‘ + — 9ac — 9bc = i9^abc , a b c. 
Let *\/a 4- *>/b 4 ~ ~ 0 : then, by the last example, 
(50 ^ + ^ + ^’ — ^s/ab — 9^ae — 9^ be" = iqS . 
a^bc 4~ b\/ ac 4- cV ab. 
Put a h c = 911 then n — ^ab — \/ ac—Vbc , or. 
