THE THEOET OF EQUATIONS OF THE FIFTH OEDEE. 
269 
then we have 
^, = 1234-2413=12H-23+34-24-41-13=A-B +23-14, 
^,=1342-3214=13+34+42- 32-21-14=B-C + 34-12, 
^3=1423-4312 = 14+42+23-43-31-12=C-A+42-13, 
^,=2143-1324=2l+14 + 43-13-32-24=A-B -23+14, 
^3=3124-1423=31+12+24-14-42-23=B-C -34+12, 
®e=4132-1243=41 + 13 + 32-12-24-43=C-A-42 + 13. 
15. We have then 
(?-(P.)(?5-^),)=(^-A+B)^-(14-23)* 
=(9-A+B)*-(7+4.1234, 
where 1234 denotes the product of the four roots; the functions A, B, C, and the pro- 
duct 1234, are each of the degree zero in the coetRcients (a, b, c, d, e); and if we put 
b = — c, 
c = — 4:06 -\-bd, 
d= 4:006— ad^— If e, 
then we in fact have 
« 2A =— b, 
«^2AB = c, 
aMBC =-d, 
a. 1234= e. 
But on the understanding that <p is ultimately to be changed into o<p, it is allowable, 
and it will be convenient to write 
2A =-b, 
2AB= c, 
ABC=-d, 
1234 = 06. 
16. I assume also 
B+C-A=a, 
C+A-B=/3, 
A+B-C=y. 
And w'e have thus 
(p — <Pi)(<p — <p 4 ) = (^-\-(^)(<p—(^)-4-4o6, and .-. also 
(<p—^2}(^ — ‘Ps)=(?>+l3)((p—y)+4:oe, 
( ‘P ~ ‘PsX?’ ~ fe) = (<? + 7 X “P “ “ ) “i" 
so that the equation in ip is 
[(P’+/5XP-7)+4a^][(p + y)(^— a)+4«e][(^ + aX<p-/3)+4ffe] = 0. 
