548 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
and 
10(be—ad) ) 
, 7 7 ( BT aaa (FESR | 
@S1(s.). HSE |b, be S..@ 5 Be —S, ). OG 
a, 5b i a, 5b | 
where a = ) = Sturm’s first function, 
g,- N=) 
= STurRM’s second function, 
bles By}T 
that is,. i ee x Sturm’s third function « Cer 
AN 
It may be shown directly that | 4, 5¢| is the proper multiplier of 8, in finding 
a, 5b 
S;. For the sextic in question, Srurm’s second function is. 
(b)(f)—(a)(9) _ 5 (U2 —ae)a*-++10(be — ad)x? ++ 10(bd — ae)? + 5 (be — af \w-+bf—ag 
a? a a ? 
and if we assume 8,=Q,.S,—S, where Q.=pa2+q, we find 
Qo = fax+5b— a oa} 
Te eats 
a, 5d 
which agrees with the former result. 
(, may be put in a more suggestive form, thus, 
Q:= 75 ap Fr {5 (ax + b).5(? — ac) —10a[2(b? —ac)a + (be—ad)]} 
a, db 
=r sar 150) | () , 5 | —(@) | @), 10(A) |} 
: Bb (4) , 5(b) (a) , 10(¢) 
= me =| (b) , B(e) , 10(d) (28). 
a, (a), 5(d) , 10(¢) 
- » (@) , BCD) 
The third set of equations for a sextic is, 
=e a 
a’,|8|0| =6 |), 5(@), 10() , 10) , (Ff) ,) » - » » | 29), 
211) Lia), dG), 20); 20d) wale) 30) 5 see 
12} 6]. , (B),5(c) ,10(@), 10%) , 5(/) , @), 
0/3) 1 So) SO) O(c) 7 L0(d) ,-5(e) 4° 
4} 6). , . , W ,5(c) , 10(d), 10) , 5(f), @) 
51 ol. . , (@ 8) ,10(c) , 10d) , 5(e) , (/) 
