154 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
and subordinate being both reckoned), will be less, by at least one, 
than it was before. Again, should the function involve a surd Y, of 
the form shown in (1), then, since s and o are unequal prime numbers, 
we may choose ¢ and w, whole numbers, such that 
est+m—wo. 
1 
gee Ua 
Let H sine when made to satisfy theconditions of Def. 8, be written K ; 
and substitute for Y, wherever it occurs in the function in any of its 
powers, the value furnished by the equation, 
1 
Ty OK)" 
1 
where it will be observed that the surd K* has no subordinates which 
were not subordinates of Y, while it has not as a subordinate 
the surd T, which was a subordinate of Y. Once more, sup- 
pose that two surds, V and V,, with the common index = are 
similarly involved in the function: that is to say, when the 
function has been arranged according to Def. 8, wherever one of the 
surds V and Vj, appears in any of its ye it occurs multiplied by a 
power of the other; as v by rae v by Vy a V ae Vi ee and so 
on ; Oe pairs of equations (3) and (4). Prop. IX, subsisting. Let 
v=" ,and V is ; and put 
- 
Al A Al 2 
a Wa 05 ) =. 
ha haz ks+8  qs+fi = 
Vics 1 mm 
BB = 
io, Vi —— WANS 3 
A being put for V* V Pe Since the surds V and Vi have the 
common index + , the expression A may be exhibited so as to involve 
only surds which are subordinates of V or Vi. Let A be so exe 
hibited. In like ca 
kel I) 
v vy, A= B ANN 
