OF THE DIFFEEEHCES OP THE EOOTS OF A G-IVEN EQUATION. 0-5 
— 2 a^cf '^ 
+ 18 a?df' 
— 30 a^ef ' 
- 10 ay^ 
— 30 d^hp 
+ 1 8 «'e/“ 
- 2 d^df' 
+ 24 a^def 
— 48 de'f 
+ 210 a^bdf- 
+ 30 a~bef~ 
+ 210 deeY 
— 66 d'deY 
+ 2 ahy- 
— 32 
— 66 crbcf' 
+ 60 dhe^f 
+ 680 a'cdf '^ 
+ 180 ddY 
— 840 d^ddf 
+ 48 (de^'f 
+ 24 abcY 
+ 2 
+ 348 a'bdef 
+ 180 a-df~ 
— 320 (dedf 
- 48 aWp 
— 52 abdej’' 
+ 264 a“bdr 
+ 48 a^be^ 
+ 120 a^edef 
-1320 d-d-ef 
+ 480 «V 
+ 348 abeeY 
+ 28 uhdf 
— 52 a-bcef 
+ 624 drdef 
— 480 a*ce® 
+ 960 
+ 60 aWef~ 
+ 624 abdY 
— 96 ac^ef' 
— 96 d^bdrf 
— 936 a'ed-f 
-1080 d^df 
— 320 ab'df ' 
+ 120 abed/' 
— 540 abed/ 
+ 720 abe* 
+ 64 aedy- 
-I- 64 a^e^df 
— 576 arcdd 
+ 1080 ard'd 
+ 290 aWdf 
— 1320 abc'f^ 
— 1596 abddf 
+ 264 aeddy 
-t- 352 
+ 648 dd'^e 
— 840 aWef^ 
+ 960 abd^ef 
— 936 addY 
— 160 ace' 
— 936 d^cdre 
+ 48 a¥f- 
— 540 ab'def 
+ 4160 abedef 
- 600 abdd 
— 48 ac'e^f 
— 192 ad'ef 
+ 432 a^d* 
— 1596 ab^cef 
+ 450 ab'd 
— 3200 abed 
— 1080 aeY 
+ 3504 aed'ef 
+ 120 ad'e^ 
28 ab^ef 
+ 210 ab'dd 
+ 960 abdef 
+ 960 abd'^f 
— 600 abd^e^ 
+ 4560 addef 
— 1728 ady 
— 32 i-ys 
— 970 ab^ce' 
— 48 ab^d'f 
+ 4560 abed'f 
— 2400 ae^d 
— 1920 aede^ 
+ 264 Weef- ! 
+ 120 ab^dre 
+ 3504 abddf 
— 4200 abedd' 
+ 960 adef 
-2880 aedy 
+ 1080 ad^d 
+ 352 wdy- 1 
4- 264 ab"cdf 
+ 2480 ab(?de 
+ 720 abde- 
—2880 addf 
-2560 add’^f 
+ 1800 aed'd 
+ 48 WeY 
- 970 wddf i 
—2160 abcdPe 
+ 2400 ade^ 
+ 1600 addd 
- 480 Wdf' 
- 576 WedY 
+ 450 Wd \ 
— 192 abc^f 
-1728 acy 
+ 480 Wf^ 
+ 960 beY‘ 
+ 450 Wdf 
+ 210 Weey 
- 936 herdf^ 
— 1440 abcd^ 
+ 960 adde 
- 600 Wcef 
-3200 Wdef 
+ 1080 6vy* 
+ 720 Wd'^ef 
+ 2480 bed-ef 
— 960 ad^e 
+ 720 def 
-2400 Wd-f 
+ 2250 Wd 
— 4200 b'edef 
— 450 Wdd 
+ 120 hc^df 
+ 640 add' 
-1920 b^cdf 
+ 2250 Wdd 
— 600 Wdef 
+ 2250 Wee^ 
+ 648 beY 
—2160 bddef 
-1400 bedd \ 
— 160 ddf 
— 450 Wed 
+ 1800 Wddf 
+ 1600 l-ed^f 
+ 2400 b'd'^f 
-1500 Wd-d 
- 960 hdy 
+ 450 
— 1400 dcde 
+ 120 ddf 
+ 800 
+ 600 b'de 
— 400 Vc'dr 
+ 1200 Wd-e 
+ 1080 h-df 
— 600 Wdde 
-1500 b‘dd 
-1000 Wedd 
+ 1200 he^e^ 
+ 960 body 
- 600 bed-d 
+ 600 hd'd 
+ 432 c'Y 
-1440 ddef 
+ 800 dd 
+ 640 dd^f 
— 400 dd'd 
In the following two Annexes, the notation of the symmetric functions is the same 
as in my “ Memoir on the Symmetric Functions of the Roots of an Equation*,” and the 
values of the symmetric functions are taken from that memoir, the powers of a being 
restored by the principle of homogeneity. The suffixes of the 2 indicate the number 
of teiins in the sum; thus in the first Annex 
the terms 23(/3y^+a7^) are equal to 2ea^]3, the complete symmetric function; the correct 
result will be obtained (though of coui’se neither of these equations is true) by writing 
23j3y^=-|2ea*i3, and so in similar cases; the insertion of the suffix to the 
2 very much facilitates the calculation, and is a check on its accuracy. 
Annex No. 1, containing the calculation of the equation n3(^— ^,) = 0, where 
— ^2=7“(7— “X 0,=af3(oi—f3), 
a, (3, y being the roots of the cubic equation (a, h, c, 1)®=0. 
We have 
V.=2)3y(0-y)=-(“-/3X(3-5'Xy-“)=?*(“. ft r)= ? \/ -z. 
where Z=27aVZ^+ &c. is the discriminant of the cubic. 
23^1^2= ^3^7 0 —y)ycc(y—a)= a(3y%y((3 —y){y—K}, 
where aQy=: — - and 
a 
237(/3 — 7)(7 — a) = 23(/3/ — a/3y — y" + ay") 
* Philosophical Transactions, vol. cxlvii. (1857) pp. 415-156. 
I 2 
