Vol. IV, No. 7.] On Rationalization of Algebraical Equations. 361 
[N.S.] 
11. Equation (7) shows that there is a depression in the 
degree of the rationalized equation if we put #=0, It can be 
easily proved that if af + / b +4 6 + aie to n terms=0, the 
degree of the rationalized equation is 2”~” 
Similar results, of course, hold for equations of the form 
bo ok 
ai +62 + c¥+ ......to°mterms=0. Here the degree of the tational- 
ized aga will be 3"~ . 
The rationalization of equations may be made to furnish 
a set of interesting identities. 
1 
1 ~ 
Thus if we rationalize #=+a?+b"* we easily find that 
(x + te"~-2a + t.e"-F.a2.........e0 0)? 
B=a(t""! + tn"-5a + ta" Fa? + .., ia sie (8) 
where é,, tg... .. are the numerical coefficients in the expansion 
of (#+a)*, 
1 
If we put eta; b= and consequently a=a?; b=£* 
equation (8) furnishes the identity 
{(a+ B)*+#,(a+B)"~-2, a2 + #,(a+P)*-*. at+....., — B*}?=a2 
{t,(a+ B)*-) + t3(a + B)*~3. a2 + cece eee}? 
for all positive integral values of n. 
As particular cases we may mention 
(1) (2@+8)*-3(a+)* . a?—283(a+B)'+3(a+ 8)? . at—G(a 
+ B)a9B3 + BS—aS =0, 
(2) (a+) ~5(a+ B)3a? + 10(a+ B)®at —28%(a+f8)§ — 10(a+ 
8)*a8 — 20(a + B)8a2B5 + 5(a + B)*a3 —10(a + B)atB*+ Blo— al? =O, 
L 
Similarly, from the equation #= at +b? +c! we get 
a — 4(a + b + c)a° + Qx*(33a2 + 23bc) —4e*{ Sa( Sa? — 2Ebo) + 4abe} + 
(a? - 23bc)?=0. 
This equation furnishes the identity, 
(a+B+y)§—diat+B+y)®. Sa? + 2(at B+ y)*(3dat+226"y’) -4(0 
+B+ y)? (2a? (Zab - 23677") + da¥pry}? + (Sat — 2287" = 0. 
Identities of this nature may be ree eae without number. 
In general, if we rationalize =a ae +......We have seen 
that the resulting equation is of degree Jma.,.,..in 2. Hence it is 
