398 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Had the two other sets of variables been at the same time transformed 
with the same determinant for modulus, we should have had the new Q 
equal to Q 7 . 
Heilermann, [H.] (1845). 
[Ueber die Verwandlung der Reihen in Kettenbriiche. Crelles Journ., 
xxxiii. pp. 174-188.] 
The determinant which here appears for the first time is different from 
but resembles those to which Sylvester’s dialytic method of elimination 
leads, being exemplified for the 4 th and 5 th orders by 
a 3 
a 2 
a 4 
a 3 
a 2 
h 
a 2 
a i 
h 
h 
a 3 
a 2 
a i 
h 
a i 
a o 
a 2 
a x 
«o 
h 
a 0 
• 
a i 
% 
• 
h o 
\ 
a o 
• 
Calling these A s , A 4 Heilermann writes his main result in the form 
% + + . . . + CL n X n __ Aq ^ ^ 
b t) + b 1 x+ ... +b m x m 6 0 
* 2 - 
A 3- 
the ending on the right being 
^2n— 3*2n'^' qj, *2m— 4*2m— 1 
^2n-l *2m-2 
according as m< or >n. 
Cayley, A. (1848, August). 
[Nouvelles recherches sur les fonctions de M. Sturm. Journ. ( de Liouville) 
de Math., xiii. pp. 269-274; or Collected Math. Papers, i. pp. 392-396.] 
Recalling his former paper on the same subject in Liouville’ s Journal, 
xi. (1846), pp. 297-299, where Sturm’s functions had been expressed in 
terms of sums of powers of the roots of the original function, he intimates 
now the discovery of more simple expressions in terms of the coefficients 
of the said function. At the same time he draws attention to the fact 
that his result may be viewed as unconnected with Sturm’s division-process, 
and it is in this general light that he prefers to state it. Beginning with 
two functions V and V' of the n th degree, namely, 
ax 11 + bx n ~ l + ax 11 + b'x n ~ l + . . . . 
v 
