DETEEMINE THE FOEM OF THE EOOTS OF ALGEBEAIC EQUATIONS. 755 
Therefore the coefficient of x r is 
and this is, by art. 2, equal to nu r in the expansion of y m in ascending powers of x. 
Hence, the lowest value of r in the expansion of the definite integral being unity, we see 
that the value of that integral will be expressed by iriyf 1 — 1), as was to be shown. 
It will be observed that the function under the sign of definite integration does not 
become infinite within the limits. Ordinary methods of approximation might there- 
fore be applied. I apprehend, however, that it is not in this direction that the value 
of such results is to be sought. 
5 H 
MDCCCLXIV. 
