266 
IMR. A. CAYLEY ON A NEW AUXILIARY EQUATION IN 
where 
and thence, obsernnff also that r-{-r'=:-? 
’ ° 'a 
a^l){ — o^fafaPfai^fco*) = — ^aJfc + ^'* + 
or as this equation may also be written, 
4:a‘'6={bac—2¥f-~^a\r—T'f ; 
and hence the Kesolvent Product 6[=fa)fa]fco^f(jJ') being determined by an equation of 
the sixth order, this is also the case with the function (r — r')®. 
5. But the twelve functions 4:(r — r') can be divided into two sets of six functions each, 
so that each set is determined by an equation of the sixth order involving a single qua- 
dratic radical. This was in fact suggested to me by Mr. Harley’s equation in t ; for in 
the case considered was =0, or ‘It—t — and the equation in t was presumably the 
particular form of the equation for in the general case. But it will presently 
appear in what manner the conclusion should have been arrived at a priori. 
6. The preceding remarks show the connexion between the function <p(=r — r') to 
Avhich belongs the new auxiliary equation, and the Kesolvent Product 6{=foofaJ^fco'^fco*). 
The relation was given for the denumerate form of the quin tic ; but taking, instead, the 
standard form (a, 5, c, (Z, e, f\v^ 1)®=0, it becomes 
4a'‘^=2500(«(7— 
7. The foregoing equation shows that ^ is a seminvariantive function of the roots. In 
fact 
fu, =x, —x^-]rcij{x^—x^)-^a)%x^—x,) +&j^(x^—x,), 
is seminvariantive, and fu*, being in like manner seminvariantive, the product 
&{=zfufu^fco^foj^) is also seminvariantive; ac — Jf and a are semin variants, and therefore <p 
is a seminvariantive function. 
8. But it is easy to show this directly. For representing, as before, the roots by 
1, 2, 3, 4, 5, we have 
(l_5X2-5)+(2-5X3-5)+(3-5)(4-5)=12-f23-f34-5(l-f22 + 23-f4)+35h 
(2_5X4-5)+(4-5Xl-5)+(l-5X3-5)=24-f41-l-13~5(2+24+2l + 3)-f35h 
and the difference of the right-hand sides is 
12+23+34-5(2 + 3) 
-24-41-13+5(4+1), 
Avhich is =12345 — 24135. So that 0 , 
=(1_.5)(2-5) + (2— 5X3-5)+(8-6X4-5)-[(2-5)(4-5)+(4-5)(l-5) + (l-6)(3 
is a function of the differences of the roots, that is, it is a seminvariantive function. 
9. To account for the division of the twelve values of +(r— r') into two sets as above, 
