152 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
and so on; where Y, is what Y becomes when T is changed into 
2;T; and P, P,, Pz, &c., are expressions which involve only surds,, 
exclusive of Y, occurring in F, (); and the whole numbers, 1, A, 
Ay, Ag, &e., are such, that, if A, , A,,,, 4... be three consecutive 
terms in the series, they are Nie to one another by the equation, 
Nao aeee Ni Neate OLN a ers! enol eae sees uly) 
For, the first equation in (11) subsists, by Cor. 4. From this we 
can deduce, by Prop. III. the following, including the first of (11) = 
B A 
NWaYo t= URINE G 
A 
y, y—'p sri, since stusjon's « ganteeptaempen deny 
B AEs 
Mad ody PMay 
and so on; where 'P is the product of an st root of unity by what 
P becomes on changing T into 2 T; 2p the product of an s*® root 
of unity by what P becomes on changing T into 2{T; and so on. 
Raise the first (2e—1) fequations in the series (13) to the followmg 
powers respectively, viz.: the first to the first power, the second to 
the A™ power, the third to the , power, the fourth to the A,” roamed 3 
pee. Sythe (e=1)™:toythe: (A "power; there™ to the (ga) 
puwen the (c+ 1)™ to the (BA¢3)" power, the (e+2)™ to the 
(8? 'Accs)™ power; ‘the’ (€413)" to” the “(6° AG ;)** power, he oe 
the (2c—2)" to the (B°7A)™ power, and the (2e—1)™ to the (G°°)™ 
power. By multiplying together the results thus obtained, we get 
AA, +B Ag+ BA Ag+ BA 2Brc-3 B(BaAc.4 t rc-2) B 
bg Gi Yo 8 A coarse Ae Cel aio aed aaah ageteren iets in 
AN KALENA Aen TONG Atos, apo 
Le Y, Ys Y3 Re ht cote cates ie AN ery Slecaeeut epee 
where P,_, is an expression like P, IP. &e., involving only such surds,, 
exclusive of Y, as occur in F, (a). - But, by (12), we have 
Ny + B = AA, 
Srey ean a 
ee ec eer eee ee 
Ne-2 tr Bro-4 aie AAG 9 
and so on; so that the equation obtained above is reduced to 
pe AAc.9 —2BAc.3 
Y Yoe aa Por Ye 
