[ 57 ° 3 
and n an odd number, it will be proper to confider 
thofe two cafes more particularly. 
i 
V. 
Firfl:, Let us fuppofe y == I, and ?i an even number: 
Then p being = i, and one of the other cofines y, r, 
j, See. = — i (Art. II.) ; we fhall have 
r J 2n 2 n V 2n ~ x + . . . . -|— 72* I) 2 = r L, z -j- o x 
V 2 4 4 XD 2 2 -}- 2yX'U-|-2-f-2yX 
<v z — 2+2 r x‘u-f-2 _ b 2r j &c. Therefore dividing 
by v 2 , 
yzn—z — inv zn ~ 3+ . . . . -\-n 2 = e v 2 — 4 ‘u 4 “ 4 x 
V 2 — i 2 qw A 2 z q*v 2 — 2--(-2rx‘U'-j- 2 ~f~2r ) 
Sec. that fadtor in which the value of the cofine y, or 
r , See. is — i, being expung’d. 
Confequently ? 2 4 is = 4 x 2 -F 2yX2-j-2rx2.-|~ 2 -S 
See. when the fadtor, whofe value is nothing, is ex- 
pung’d. 
VI. 
Let us now fuppofe y = — i, and n an odd num- 
ber: Then one of the coiines y>, y, r, 6 cc being 
= — l (Art. III.), 
v 2 " — . w . . ~f n 2 v z will be = ^ -f-o 
Y.V 2 2-f- 2pXV-\- 2- J [-2pX'V 2 2 -j- 2 y X'U-f- 2 -f- 2 y, 
&c. Therefore, dividing by t 4 , 
v 2n - 2 __ 2n <y 2# 3 -f n 2 will be = 
V 2 — 2 + 2 /> x n J -j- 2 -f- 2p'* t V 2 — 2-j-2yX‘U+ 2 -f - 2 y, 
See. and confequently 7i 2 — 2 -\- 2 px.i-\- 2 qx 2 -\- 2 r > 
tee. when the fadtor, whefe value is nothing, is ex- 
pung’d. VII. 
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