Applications of AheVs Theorem. 121 



tan' -I' tan' I?- = 1 



<^l + ^2 = ""• 



I shall now consider the integral 

 p dx 

 J -v/l+a:"' 

 which cannot, except in certain cases, be reduced to an elliptic 

 integral. 

 Suppose 



^/\■\■x^ = 's/'^x^ i+a;" = 4^1 or, ^,^x = 1 

 fi a: = 1, ^2^ = 1+^1 ^ • 



1 +;r»- (I +c^xY = X (;r«-'- £!lf2'3Y-^^Vl^ 



+ a^j a:*2 ^3 ... ^n— 1}> 



the upper sign to be taken if n is an uneven number, and 

 the lower if jris even, x^^ x^^ x^ ... Xnr-\ being the roots of 

 the equation ^n-i _ Pa; + Q = 



r> **i -^a «^3 ••••*' n— .1 



/. - ^ , 



together with the other conditions implied by the nature of 

 the equation ^.«-i _ p^ 4. Q == o 



^1 - + 2 • 



a*!, jTg, 0^3 ... a"„_i, being subject to the above condition 



«>y^i^ + nhj^ + .^. /^:!ii^ 



*^0'/l+*" •^0^1+'*"* ^o'/l+*» 



= constant. 

 Also, 



s^P^^M^ + ../:i££f + .^^.ffmlJ^ 



= constant. 

 K coefficient of — in the development of 



L \/l+A"j 



according to descending powers of x. 



Third Series, Vol. 6. No. 32. Pe^. 1835. R 



