Section III, 1921 [41] Trans. R.S.C. 



On the Reduction of the Ciradants to Polynomial Form 



By J. C. Glashan, F.R.S.C. 



(Read May Meeting, 1921) 



Notation. — (i) co = a primitive root of x" — 1=0. 



(ii) P;, = coV + û^'S + 'o-''S3+ +co^"-^)^>„_i 



(iii) ai= J-p/ + p,' + p3^+ +p|j 



(iv) C{n) = (.T + pi) {x-\-p2) (x + ps) C'v+pJ. 



= .A;" + r, x"--^~cs .t"-^+ (-1)% 



(v) eo+ei + e2+ +e„_i = « conditioned by 



ei+2€2 + 3e3+ +(« — l)e„„i = mod n. 



(vi) m = n — eo. 



(vii) €i,^ + e2,;, + e3,^+. +e„_i,^ = m^ conditioned by 



61,^ + 262,^ + 363,;,+ +(«-l)e„_i,^ = mod n 



M = l, 2, 3, / 



and €x,i + 6x,2 + ex,3+ +ex,/ = ex 



X = l, 2, 3, n-1 



«?i + W2+ +W^ = W. 



(vni) P^= ^ j-^î ^ 



6l, ^ • 62, ;i • ^n ~\i y.- 



(ix) I >'i''3'2V3'^ 3'm-i|= the sum of the non-equivalent pro- 

 ducts among j'^ 3'?2x' ^W y\n-i)\' in which X = l,2,3, . .n-l 



in succession and '/X' = the least positive residue of /X mod n. 



Examples. For n = 7 



\yiy&\ =yiy&+y2yh+yiyi 

 \yi^yh\=yiyb-\-y2^yz+yzyi+yi-y&+yhyi+yiyi 

 biy^yil =yiyiyi-\-yzy&yh 



(x) [0^0 lex 2^2 (n-i)6'- T = s|:.«-^ y'l-p'y'Lp y'::zl-p'\ 



in which p = 0, 1, 2, w — 1 and 'a — /?' = the least positive 



residue oi a — p mod «. 



Examples.- — ^For eo = 3, ei = 2, €2 = 1, €3 = 1, 64 = 0, €5 = and e6 = 0, 



[0^ V 23]= x^\yi-y2y3\+x^yiy2y6^\-\-x\yiyi^ye^-{-x\yi^y5'^ye\ 

 + |3'3'''3'4-3'53'6 1 + b'2^3'3-3'43'5 1 + l3'ib'2->'33'4 1 

 For €0 = 3, 61 = 3, 6„ = 0, 63 = 2, 64 = 65 = 66 = 0, 67 = 2, 68 = 69 = and 

 €10=1 and J = 10 



