1919.] On the Rationalisation of Algebraic Equations, 307 
and their products taken 2, 3, etc. at a time. Thus the right- 
hand side of the equation consists of 
1+{(J-1) +(m—-1)+....... } 
*(1=—1)m=—1) 4 ee } 
+ (L—1)(m—-1)....... (n—1 
terms, i i.e. (l.m.n. te Therefore in this case also the 
. ) terms. 
previous method is s effective in giving the rationalised equation 
of the (l.m.n. ....)th degree. 
In case I, the degree of the equation cannot be less than 
n, for i 
1 
v=f (p") 
is a root, so will also be 
1 
a=f (6p") 
where @=1. 
In case IJ, the degree of the equation will be m’, for if 
rene 
z=f(p", 7") 
is a root, so will also be 
1 1 
: z=f(0,p" , 629”) » 
where 6,"=1, oF ech. 
In case III, hes different roots are similarly given by the 
ASB Mss Ss expressions 
=f (0, p : 0,0", 0, -++) 
where 6, =1, 9, =1, 6,"=1, 
and thus the degree of the equation in this case will certainly 
Lmn.... 
oN eee eS 
