Vol. IV, No. 7.] On Rationalization of Algebraical Equations. 359 
[N.S] | 3 
If, however, we had rationalized the equation z= at + bs first, 
the result might easily be expressed as a determinant of the 5th 
order and so on. 
There are two cases of equation (4) which can be rational- 
ized by very elementary methods without the help of determin- 
ants, vtz., 
(1) when l=m=n=.,,.....=2 
(2) when l=m=n=......=3. 
For [=m eS lage, 
a i 
fle-e* )=fitfe. oF =0, 
A * of, =0. 
Again, when l=m=n= =3 
we have f(a- O=fth: if. c8=0 
2 
fro +f aah, 
cubing both the sides, we get 
8 3 i & 8 
fo .ct+fz -+3f, .c% xfg.c5x —fp=—-f, 
8 8 3 
whence fi +e. fe +e. fg —3ef, - fy -fg=0. 
We can thus rationalize any equation of the form~.  _ | 
2 4b 408 4 
e=a?+b? +e iS owes 
She igs } (B) 
ee t= =a? +b hee 
by anit elementary methods and without any Iniatrisdge of deter-- 
mina 
The same methods would also a for the more general case 
1 
when a= ffa* 3, 8, &e. )+ (18, m* , #°; &e.) where i, ¢ are 
rational algebraic functions. ; 
We e might also rationalize ae of the = 
ee oe 
2=f(a' tb”, co”, &e.), 
where 1, m, n- are of the form 2” . 3° (p, g being integers or 0) 
by means of suitable substitutions. 
But the results are not, in general, obtained in wists lowest 
terms and the method is practically useless, 
. ss 
Equations of the frat gaaee aan veseee Where rr ae -3 
can, however, be rationalized by means of elementary. methods and 
results ohestned in mae lowest farms. : 
