DETEBMINE THE FOEM OF THE BOOTS OF ALGEBEAIC EQUATIONS. 747 
on interpreting the T functions in the usual manner. We may therefore write 
n—l _i_ 
u=u n -\-%Qi f f dsdte~ (s+t) s» _1 1 ~ » _1 f e“ ih dh, 
Jo Jo Jo 
since by the symbolical form of Taylor’s theorem 
tt-i-p 5 / i_\d / n-i _i\ 
s n t n <p(x) = [s n t n ) <p(x)=<p[xs n t n j. 
Let us now transform the double integral relative to s and t by assuming 
s=vt, 
and making v and t the new system of variables. We shall have 
dsdt=tdvdt , 
while the limits of v and t will be 0 and oo. Hence 
u=u 0 - |-£C £ f f dvdte~ (1+v)1 v n e* ih dh. 
Jo Jo Jo 
Again, transform the double integral relative to t and Ji , by assuming h=ty. We shall 
have dh=tdy, and the limits of y will be 0 and xv~ . Whence 
u=u 0 - HSCjf f f dvdtdye~ l ' l+ 
Jo Jo Jo 
Integrating with respect to t, we have 
m j 
u=u 0 -\-%C\ dv I dy — - . 
JO Jo 1+V — UiV 
-1 . 
Now integrating with respect to y , and merging — in the arbitrary constants, 
w=w 0 +2 dv ^ _1 |log^l+v— log(l+w)| 
,+2cJ dw^logn-^— 
( 4 ) 
It remains to determine u 0 . 
Developing the function under the sign of integration in ascending powers of x, and 
effecting the integration for each term separately, we find, for the coefficient of x n , the 
expression 
Un =m An — — nj 
wr(ra) ’ 
5 G 
MDCCCLXIV. 
