EEV. E. HAELET ON THE METHOD OE SYMAIETEIC PEODHCTS. 
345 
17. In order to facilitate operations it will be convenient to replace the coefficient of 
the first term by unity, and to suppose the qumtic to be deprived of its second term ; 
that is, in efiect, to deal with the equation under the form 
(1, 0, c, d, l/=0. 
Then, if we assume 
where, as in former articles, u is an unreal fifth root of unity, the following relation 
will obtain, viz. 
r being 0, 1, 2, 3, or 4 indifferently. Developmg and reducing by the known properties 
of CO, we find 
2a-^=10(/3,/3,+/3,/33), 
=15 (/Si/Sg + f3^f3i + f3^^2 ? 
2a;^==30(/3-f/3!+/3®+20(/3?/3,+i3^/34+/3,^/33+p^^ 
2r=52/3^+100(/3’/33i34+i3.^i3,i33+/3!^^^ 
+ 150(/3m+/3m+/3:i3ii3i+i3m)- 
And by the method of the limiting equation or otherwise, 
2a:^= — 2c*, 2a'^= — 3c7, XT^=2(f—4e, XT^=5cd—5f. 
Whence by comparison with the above. 
— c ( 4 . 2 ^ 3 ), 
— cZ= 5 (/ 3 f/ 33 + ^3/3 1 +f 34^2 4-/33/34), 
-^=5(/3?/32-b/3p.-h/3^/33-h/33^/30-f3/3./33^3/34-^^^^ 
-/=2/3^-10(^?/33/34+/3^i3,i33-H/3^/33/3,-ff/3m)+i^^ 
= 2/3® + 1 0 (/3i/ 32/34 + + /33/3 j/32 ) — ^cd *. 
18. If now we suppose one of the constituents, say to vanish, and eliminate the 
remaining ones, the effect will be the same as if w^e supposed the symmetric product, 
expressed as a function of the coefficients, to vanish. For 
5^/31/32/33/34 = ^; 
* The formulse here exhibited are Evleb’s, or rather Evlee’s as simj^lified by Mr. Cockle. They indi- 
cate the connexion between the coefiBcients of the cjuintic and the constituents of its roots. Those consti- 
tuents, by Lageakoe’s process, are expressed as rational functions of the roots. It is to be observed that 
Eviee’s functions are cyclical. In fact, applying 2' to the cycle 
... 12431243 ... 
the relations may be written thus : — 
-2c=51'i3,/3„ -d=5S'/3%, -e=52'l3^,r^2+^(3f,2f^3f34-io\ 
-f = Z(3^~10^'l3^,(3j3,+icd= 2/3^ + (3lf3,(3,~^cd. 
And by the working properties of 2', 
cd= 5^2.'(3^,ft3(fi,l3,+ ( 32 ( 3 . 3 ) =5^2'(/3“A,/34+/3?/3^/34). 
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