TRANSACTIONS OF THE SECTIONS. 13 



Theorems on the n^^ roots of Unity. By J. W. L. Glaisher, M.A., F.M.S. 



If n he any nimiber and if all the sets of r elements that can he formed from the 

 numerals 1, 2, 3 . . . w — 1 be written down according; to any rule with regard to 

 sequences and breaks, then the sum of all these sets will always be rational if the 

 numerals 1, 2, 3 . . . w — 1 be supposed to stand for 1— .r, 1— .r-, 1 — x^... 1—.%"-^, 

 X being any prime «th root of unity. 



To make the theorem clear, consider an example. Take n= 7, and write down 

 all the sets which can be formed from the numerals 1, 2, 3, 4, 5, having (say) a 

 sequence of two and one break (». e. having two numbers consecutive and one non- 

 consecutive or isolated), viz. these are 



124, 125, 126, 235, 236, 346, 341, 451, 452, 561, 562, 563 ; 



then the theorem asserts that x being a 7th root of unity, the expression 



1 -X . 1—x^ . 1—x* + 1 -.r . l-.r= . 1—x^ + 1— .r . 1— .r^ . 1— .r«+ . , . 

 + 1— .r^' . 1—x'^ . 1— .r^ + 1— .rM— .r" . 1—z^ 



is rational. The simplest case is that of a sequence without any brea k ; ex. gr. con- 

 sider a sequence of two, then, since 12, 23, 34, 45, 56 are the only sets, the theorem 

 asserts that 



l-x.l-x'-\-l—x^.\-x^-\-l—x\l~x^-\-l-x'.l~x' + \~xW-x^ 



is rational. 



The general mode of proof will be easily gathered from the demonstration of the 

 truth ot the theorem in the case of these two examples. Take the second first, and 

 consider the function of s, 



(1-=) {l-xz), = l + Kz+-Qz\ say, 



X being a 7th root of unity. Since 7 is a prime number every root is a prime root, 

 and the roots of the equation .r'' = l are x, x-, x^, x*, x', x^, 1 ; so that, substituting 

 successively these values for s, and adding the results, we see that 



0—x)(l-x.x) + {l-x''){\-x.x^)+(l-x'')(l-x.x^)+{\-x%\-x.x') 

 + {l-x')(\-x .x') + l\-x'){l-x .x^)+{l-x'')0.-x .x') 



(the last two terms being zero) is rational, since the coefficients of A and B vanish 

 by the summation. 



To prove the theorem in the case of the first example, note that all the sets may 

 be obtained by starting with the three in which the sequence is 12, and continually 

 adding imity to each of the three numbers in each set, thus : 



(in which 8 is replaced by 1 as it arises). We anive after a cycle of seven lines at 

 the original line again ; and, ignoring the terms in which 7 occurs, since 1 — a;'' = 0, 

 we see that (1, 2, 3 . . . denoting, asstated above, 1—^, \—x'^, 1—x'^ • • • ) the first 

 column, viz. 



124 -f- 235 + 346 -f 5G1, 



is formed by putting s equal successively to x, x^, x^, .r', a;', ,r^, 1 in the expression 

 1—z . l—.xz . l—x^z, which in of the form l+A=-}-B8^-|-C«^; and similarly for 



