422 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Brioschi, F. (1854, February). 
[Sur les fonctions de Sturm. Nouv. Annales de Math., xiii. pp. 71-80 ; 
or Opere Mat., v. pp. 89-97.] 
Brioschi in effect here recalls that if /, f x , f 2 , . . . . be the series of 
Sturm’s functions originating in the consideration of the equation 
x n + ape*- 1 + a 2 x n ~ 2 + . . . . + a H = 0, or, say, f(x) = 0 , 
and q x , q 2 , ... . be the linear functions of x which are the quotients 
obtained in the process of finding f 2 , f 3 , . . . . then 
(!) /=?l/l-/ 2 > fl = <hA~fz> > fr -2 = q r -Jr-l-fr- 
(2) f r is of the (n — r) th degree in x. 
(3) From (1)4=1 I 1 
/ ■ ■ ■ 
(4) The successive convergents 
fraction being N 1 /D 1 , N 2 /D 5 
?2 
<h’ M2- 1 ’ ' 
to this continued 
II 
5- 
q 2 l 
II 
O' 
ffl 1 
l & l .... 
1 q 2 1 .... 
. 1 5 4 ... • 
. 1 q 3 ... . 
Vr 

(5) From (1) after eliminating / 2 ,/ 3 , . . . , f r _ x by repeated substitu- 
tion or otherwise 
fr = 
(6) From (1) after eliminating/^,/^, 
fr 
f = D r-lfr-1 - B r _ 2 /r . 
With these facts before him he seeks to find expressions for f r , D r , N r , or, 
say, for the coefficients in 
A r> 1 x n ~ r + k ^- 1 + • • • , 
B Vil x r + B r> 2 x r ~ 1 + • • • , 
C r iX r 1 + C r 2 x r 2 + . . . , 
failing to note that, Cayley having in 1846 found such an expression for 
Sylvester’s substitute for f r , the annexure of a known multiplier to 
Cayley’s result would have given him the most important of the three 
expressions sought. 
In the first place he deduces from (4) that the coefficient of the highest 
