( ^12 ) 



further, from llie LP^AK,, l\AK,, l\AK, wo get 



dg (p ct(j (f) 



cos i If' COS If? 



while (Hrect from llie fi,<>ure we get 



(•^) 



Xn = .T - ., VN X4, = -T - 2.f', X„ = rr - ^ if<. 



(0) 



Xsi =^— ., V'- X5, 



l«l'. X.. = ^ - ., 'f- 



If we \vi-ite the priiieiph' of ratioiuil iiKlices 



«^41 ''.s '^.. 



— ^1 • ~2 • ~a' 



(7) 



;iii(l (»l»ser\'e lluil in the quotients of eaeh two (hu^^> siiiec 



I (Si,i z=i cos {K,l'i,) = r,>s y,ro.s' </ ( 1 + t;l (f t>i y, ros yj,;) 



the fuctor standing before the l)rackets ahvavs eaneel, tlien we euii 

 easilv introduce the vahies (5) and (Gj and obtain from (7; 



15 17 



OO^ ^- IjJ — con ^ If? . Q . (^OS ^ If? — COS ^ If 



2 2 cos 1? — cos 3i? 2 2 



1 3 



cos — if? COS — tf^ 



Tliis gives (Hreclly 



cos If' — cos 2 if' 1 5 



cos — if? — cos - if; 



2 ' 2 ' 



l • ~i • ~3" 



8 



.sin — if' . ^ ■ ■ f^ 



2 .sm 2if? .s/« if> 5«n 2if? 



^i * 3 1 ' sin If? 



shi ~ If' ^/// _^ If' --'ii -^ If' 



'1 • "3 • ~Z 



(8) 



If we now take tlie lirst and hist members of this (h)ul)le [tropor- 

 lion we have 



3 

 sin - If' 



sin - if? 



sin 2if? 

 sin if? 



i. e. equal to a rational fraetion. (»r also 



1 + 2 cos M' _ • . _ ^ _ .' 



2 cm If' — ' • • '"■' "~2(7' — 1)~ 



where r' is also rational. 



This i-cfjuirement, when if' = 2.t // and /; > 5 is fulfilled only 



tor n = (3. 



