358 Journal of the Asiatic Society of Bengal. [July, 1908. 
soear this can be immediately rationalized by the method explained 
So tha t, if we can rationalize an equation of the form (4) when 
there are p terms on thé right-hand side, we can obtain the oa 
when one more term is added, in the form of a determinant. ‘Th 
method of rationalizing equations of this form is thus obvious. 
It is further to be observed, that if f(a) be of the kth degree 
1 rt = 
in # and f(a—c") be expressed in the form $,+c".¢,+c¢" . $g 
n=-1 ; Eve 
Pisses c ” . $a, $, must be of the kth degree in za. 
"It is clear from equation (3) of Art. 6 that the degree of 
the rationalized equation, when the term c” is added, must be k x n. 
1 
Now if oe = rationalized equation is #! =a 
if = S a the degree of the rationalized equation must 
Leaiea and so o ey 
Consequently when a=a' +b" +e" +....... , the degree of the 
rationalized equation in z must belxmx~7...... 
This is, of course, otherwise evident from a known property 
of eliminants. 
S; To illustrate the method of Art. 7, let us take the equa- 
tion, =a? 248 408, 
If ea ot: we have #'=c, 
L 1 i 
eer 8 - wmat+c®; (ez—a?)b=c 
eh ea 1 1 2 
or x’ —52*a? + 1028a ~ 10x20 . seeded ak atae 
be ! 
or (a5 + 10x8a + 52a®—c) — a?(5a*+ 10z°a + 0%) =0 
(a5 + 10z°a + Sava? —c)?—a (5a + LOzta + a2)? =0 
or 210 — 5a8a + 1025a? — 2cx’ — 10n*a? — 2025 ac + 5a2a* 
— 10za%e + c?—ab=0, 
This is our f(#); if we onpane fla-bh) in the form 
fiths - BS +f bs =0. 
The rationalized equation will be 
fy fy 
fs 
bfs, hi fa 
bfz, (dfs, fi 
=0 orf; * + bfy + + bf, — Bf; fa f= 
which is evidently of the 30th tite 



