But if we make k=8, or take for n, we have 1, ar=en=as, as=als=a1, so that we get no roots from this series but a', I, which are only the three cube roots of 1 (cube roots are among twelfth roots). But choose (5 is prime to 12) and its successive powers are aw, a12 or or on, or e, or ag, or or a', or a, or or or 1, after which the series recurs in the game order.
4. If m be any factor of q, all the mth roots of unity are among the qth roots. Thus, if q va=n, and if a be the first of the series of qth roots, the /nth roots are a', (4'0 or 1. For (a' )" =ae &c. All those powers of a which have exponents prime to I, may be called primary qth roots of unity : thus the primary 12th roots are a, a".
A The qth roots of unity exist iu paint of the form cos pt sin's ‘I(- 11. Those pairs are a and at and at ore and and • &a.
Let the question now be to find the qth roots of -1. If we now take -1 semi Or +21-s) +sin (e + 2hr) . a/(-1) we have all the 9th roots in the distinct values of the formula (21• +11a (21+1)ir -113 9 +ain 9 - .
Let a ...cos + sin a/(- I), then the qth roots required are a, t 9 eginning with k= 0, and ending with k = q - 1. Thus, if a be any one root, all the odd powers of (positive or negative) are also roots, but do not contain among them all the mote unless the value of 2k+1, from which s is derived, be prime to I. Thus if qs.15,
and if a es, we have (since sl'.21) eS = al that we only get, from the powers of the distinct roots e, a", -1, al, eM, which are also the fifth roots of -1. But if 2k +1 be prime to 9, all the 9th roots of -I may be obtained from a. And if as he any factor of q with an odd quotient, all the rah roots of -I are among the 9th root. Also these qth roots occur in pairs of the form cos itt ± sin 9)v( -1), the pairs being a and es-1, end he., or a and as and &c.
Emery 9th root of -1 is one of the (2q)th roots of +1, and the t29)th roots of +1 consist of all the 9th roots of -1 and all the 9th roots of + 1.
The following equations will also be (wily proved ; =cos 4)1. +sin (2k+ 1)r (2k +Pr (2k + I) - V(-1)}' + sin As it is not our object here to write on the applications of these forrimhe, but only to supply an article of reference for those who may have forgotten or imperfectly learnt the groundwork of this very important branch of analysis, we finish here, referring to Seems for such applications as fall within the plan of this work.