544 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
and also = {(aB —ay)(yi—fS).aByd; : ‘ (14), 
which appears to reduce to, —P,2 {(a—f)’yd}. This, however, containing only 
‘ ; 6.5 
6 terms, is but part of the true sum, which contains fee = is terms. 
On expansion, we find 
= {(aB—ay)(y8— 88); = — 22 {(a—B)y8; — 2 (aB—7)"5 
the 15 terms being, a set of six terms in = {(a—)?y6} twice over, and a set of 
three squares in > {(aB—y-y6)*}. 
Now, Bs ys Ks Ae agp: rele > {(a — B)*y8} 
i ihe Sol bys, ays , od , ay 
=P,P,—16P, 
and also ‘i P,, 4P, 
Ps 
Hence, dividing by P, (which, but for symmetry, might have been left out from 
the first), we have 
(a8 —y8)' + (ay — 88)° + (a3 — By)’ =P}—2P,P, —4P,P, 
36c2 — 32bd — 4ae = 
a a : (15). 
Cor. 1. Hence, from results in IT., 
40°X {(a8 — yd) = 144( — bd) + 16(bd — ae) 
=a'2 {(a—B)(y +8)" 
verifying the above equation. 
Cor, 2. One of MActaurin’s conditions for imaginary roots is 
P? <2P,P, +4P,P, 
or, in (a’, Be’, 6 ay AY, 9. 020d 4+ Aale oe CM 
VI. The results arrived at may, in the case of a sextic, be written in the 
following forms : * 
* T have to thank Professor Cayituy for valuable suggestions, in accordance with which the nota- 
tion on the left hand sides of these equations was made to harmonise with that on the right hand sides. 
This will be seen more fully in the next section. 
