268 Proceedings of Royal Society of Edinburgh. [sess. 
Coming now to the case of S 3 = 0 we proceed differently, ths 
three roots being localised by observing the changes of sign in S 3 
as we pass from one value of the variable s to another. Four 
values of s which suffice for the purpose are - oo , s 4 , s 2 , + oo . 
No reasoning is necessary to show that, when s is = — oo , S 3 is 
positive, and when s = + co , S 3 is negative. When s = s 4 we have 
5 2 = 0 , and therefore know from our auxiliary theorem that S 4 and 
5 3 must have different signs, — a fact from which we deduce that 
S 3 is then negative. Similarly, when s = s 2 , it is seen that S 3 is 
positive. We thus have the set of values 
s = - oo } s 1 , s 2 } + oo , 
and S 3 = + j — , + , + , 
which shows that one value of s which makes S 3 = 0 lies between 
- oo and s 1 , another between Sj and s 2 , and the third between 
s 2 and + oo . In other words, the roots s', s", s'" of S 3 = 0 are 
such that between each consecutive two of them there lies a root 
of S 2 = 0. 
The case of S 4 = 0 is treated similarly, the five values given to s 
in S 4 being 
/ rr rn 
-CO , S , S , S , +00. 
As before, there is no difficulty about the first and last of these, the 
value of S 4 being seen to be positive for both. When s is put =s' 
we know that S 3 vanishes, and that therefore S 2 and S 4 must have 
different signs. The sign of S 2 is settled from recalling that s' lies 
between — co and s 4 , and that for these values of the variable S 2 
is equal + oo and 0 respectively : consequently the putting of s = s' 
makes S 4 negative. Similar reasoning enables us to complete the set 
S = -co , s' , s" , s'" , + CO 1 
S 4 = + , +i 
from which we learn that one value of s which makes S 4 = 0 lies 
between - co and s', a second between s' and s", a third between s" 
and s'", and the fourth between s'" and — co . 
Having reached this point Cauchy adds — 
(t Les memes raisonnements, successivement etendus au 
cas ou la fonction s renfermerait cinq, six, . . . , variables, 
fourniront evidemment la proposition suivante : 
