536 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
and also’ = | Py y, 223238, —SPero 
0? 
that is 48(b?—ac)=a?.2{(a—B)*} . ae 
Again, the symbol 
pee | Btyt+8 , atyt8 , at B+6 , a+ Pty | 
1,1,1,1] Byt+fd+y8, ay+ad+y6, aB+ad+B5 , aB+ay+ By 
= ie B+y +8 , aty+68 Wee y+é, y 
1 | By+Bd+y8 , ay+ad+ yd yo fe. 
and also = ES , 3P, | = 4P} — 9P,P, 
ree 
2 
that is ~  -144(c? — bd) = a?. 2{ (a— B)*(y + 8—y8)} é . 2) 
Likewise, 
la, 8s¥50 AeA eiba mien kt puede are 
1,4;1, 1) Byer rio ane creer 
an eee De eae ay +ad + yd Pie 2{(a—f)? i ae SP 
An ’ ays ? 
and also a 3P, , 4P, | = 3P;- 8P,P, 
(seb os P, 
that is 48 (d?—ce) =a?. 2{(a—B)? yd - j : . to 
The last result might have been obtained from that for b’—ac by consider- 
: ie aes eee ; ; : bens, 
ing the roots |» By’ § in which case it would have appeared in the form 
3 { (ay3— By8)'} 
In an equation of the fifth degree the results are similar. 
[For an equation of the mth degree, we have always n*(x —1)(?—ac) 
=a’. (a—)"t]. 
As an example from the fifth degree, take the following case. 
The symbol 
a € | 
rita 
yo+yet+ Se , aytad+acetyityetSe,..-. cee pees Fi 
By8-+Bye+ Ade-+ Be. 5 ab + ore-tabett ys | ssa 
