Vol. IV, No. 7.] On Rationalization of Algebraical Equations. 357 
[N.S.] 
It is evident that equation (1) can be written in the form 

Prt bey t by? + ever + omy" 1=0 (2) 
where 9), $2, «+... $m are rational functions. 
Multiplying (2) by 9. nies y"-1 and observing that y™=a 
we get the equations: 
Abm+9)y + Poy? + wre. + Om-1y"-1=0 
Abm-1+ AP mY #OY? + ...00. +m-2y"-1=0 
A9m-2+ AP m—-1y + apmy?+ seavee + $m-3y"-1=0 


ab,+abz,yt+agdy.y?t+...... + oy"-!=0, 
ring together with equation (2), form a system of m 
equation 


jirasiuies 91 Hs 9? s0000 y™-1 from these m equations, we get 
i, 2 3 Pm 
aPms 91, 4 Po) Ym 1 
a?m-1; abm: 9; seeeeereness Pm-2 
APm-25 APm—1y APmy sess ?n-31=0 (3) 





AD, Ay, APyyrresescereeeeP} 
Thus, equation (1) can always be rationalized and the result 
corse in the form of a determinant of the » th order. 
e may observe, in passing, that when a=1, the determinant 
(3) reduces to a very familiar one, which possesses inter resting 
properties (see Arts. 23—25, Chapter viii, Scott and Mathews’ 
‘Theory of Determinants). 
ee ee 
7. When BHA ED HC” Hccevever ene (4) 
Let f(«)=0 be the rationalized equation when one of the terms 
1 
om the right-hand side, say ee is left out. 
1 ~— 
If ae rege. ak, See eee 
it is ype that Ay) = =0 is rational in a, },....... Hence 
f(@ a )=0 is rational in a, b, ...... : Now if f(x) be an algebraical 
function, f(w—c") must be of the’form 
ie8 n-1 
Pit G20" + Pye" +... Hc * =O, 
