Hh PRINCIPLES OF THE SOLUTION OF EQUATIONS 
1 1 
(d — 4,) + dy 4)" + dy 4,” + ete. = 0 (48) 
1 
1 
ae ‘ 
are separately zero. Therefore (d + d, 4 a + ete.) = ¢ ae 
t, being a primitive m* root of unity, 
1 1 
== ’ 
_ ; / 
(d + dy ty 4 +ete)™"=dt+dt 4 + etc. = d; 
Therefore, 
1 1 2 
(d+ dt) 4," + ete.) = tf (d +d, 4," +d, 4," + ete), 
aie : ; ; ‘ 
t; being one of the mt‘ roots of unity. In the same way in which 
1 
the coeflicients of the different powers of A in (48) are separately 
zero, each of the expressions d (1 — tt ) di (4 — a ), ete., must be 
zero. But not more than one of the m — 1 factors, 4 — oe 
2 
yo i. etc., cam be zero. Therefore not more than one of the 
m — 1 terms d,, dz, etc., is distinct from zero. Suppose if possible 
1 
that all these terms are zero. Then ¢ As = d, Therefore the 
1 
m : . 
different powers of 4° can be expressed in terms of the surds in- 
1 2 
volved in d and of the m*® root of unity. Substitute for ” , 4.” 
ete., in (47), their values thus obtained. Then (47) becomes 
moe mt! 
Q.— (hy 4 eee. a) ee (49) 
where @ involves no surds, distinct from the primitive m*® root 
1 
. ™ : 
of unity, that are not lower in rank than 4, ; which, because 
1 
the coefficient of the first power of A in (49) is not zero, is, by §8, 
impcssible. Hence there must be one, while at the same there can be 
only one of the m — | terms, d1, dz, ete., distinct from zero. Let 
