OF THE HIGHER DEGREES, WITH APPLICATIONS. 113 
d, be the term that is not zero. Then ¢ — {{ = 0. Therefore 
1 — js not zero. Therefore d= 0. Therefore, putting p for d, , 
1 c 
“m “m 
t 4, =S P 4, 
§50. Cor. By the proposition, values of the different powers of 
1 
a can be obtained as follows ; 
= vee a, 
ae A aS. fee 
=¢ 4, »bA =k A 
|e 
m “m m 
as Day , 
td =pd4, ,t4, etc.; (50) 
where 7, q, etc., involve only surds that occur in 4) or 4, ; and ¢, s, 2, 
ete., are whole numbers in the series 1, 2, ....,m— 1. No two of 
the numbers c, s, etc., can be the same ; for they are the products, 
with multiples of the prime number m left out, of the terms in the 
series 1, 2, ...., — 1, by the whole number ¢ which is less than 
m. Therefore the series c, s, z, etc., is the series 1, 2, ...., m— 1, 
in a certain order. 
§51. Proposition XVI. If 7, be one of the particular ccguate 
forms of /, the expressions 
1 iat m —2 m—1 
m m m ™m 
TS eal AM a OP. aT 1) 
are severally equal, in some order, to those in (39), ¢ being one of the 
mh yoots of unity. 
By §47, one of the terms in (42) is equal to 7}. For our argument 
it is immaterial which be chosen. Let ra = 7}. By Cor. Prop. 
XYV., the equations (50) subsist. Substitute in (47) the values of the 
1 
different powers of 4” so obtained. Then 
(t—1 p Am + t—? Gay Aa + etc.) 
me nai 
— (4," +4," + etc) =0. (52) 
sa is 
By Cor. Prop. XV., the series ae ; Aw , ete., is identical, in some 
122 ee eit, 
order, with the series Ay 4.” , ete. Also, by §8, siuce 4 aide Ae 
ip ley 3 y: 1 
