REPORT ON CERTAIN BRANCHES OF ANALYSIS. 281 



p-^d^^ _^=.log {P-P- + 9,p ^Th) + const.; 



and if we denote by r and r' the extreme values of p, when a: 

 = — 1 and X — + 1, we shall find, 



r + ^dx _ 1 ^ 2r\/ab + ^ab-{a + h){\ + ah )'} 



^~J -\ "y ~ 4 s/ah ^^i2r^ //ah-^ah- (a + 6)(l+a6)y 



inasmuch as^-^ is 4 a 6 — (« + A) (1 + « 6) in one case, and 

 a X 



— A^ah — {a + h){\ + « 6) in the other. It will appear like- 

 wise that r and r' will have the same sign, whether + or — , 

 in as much as p will preserve the same sign throughout the whole 

 course of the integration. If, therefore, r' = + (1 + «) (1 + a), 

 then r = + (1 - a) (1 - 6) ; and if / = - (1 + a) (1 + h\ 

 then r = — (1 — a) (1 — h). It thus appears that (1 — «) (1 — h) 

 must have the same sign with (1 + a) (1 + 6), and consequently 

 if a 7 1, and h 7 1, we shall have, 



. = _L^ loff (<?-!) (^> - 1) j/^ + 4a 6 - (a 4- 6) (1 + «6) 

 4 ./a 6 (a + l)(6 + l) v'"«6-4a6-(« + 6)(l +«*) 



= - — =- log . Iv «_ + _j_ (stj.i]jinrr out the common divisor 



_ 1 - a/« 6 + 1 , 



2^ah-a-h)^ ^^^ log ^^T^^^^Y = ± -4- 



If a 2: 1 and 6 l\, we shall find r = (1 - a) (1 - h), and 



« = === log I 7= ) = ± «i. 



2 A/a 6 ^ Vl - A/a 6/ 



If o Z 1 and h 7 1, we shall find r = (1 — a) (1 — h), and 



1 , ( Vh + a/«^ 



X = 



1 ( Vh + A/a\ , 



log ( ~T ^ ) = + ^2. 



2 -/afi 

 If a 7 1 and 6 Z 1, we shall find r = (a — 1) (1 — h), and 





2 \/ 



It would thus appear that the definite integral would furnish 

 erroneous values of z if no attention was paid to those values 

 of the factors of r and »•', which the circumstances of the inte- 

 gration require : and it may be very easily shown that an atten- 

 tion to the developements of K and K' will, with equal certainty, 

 enable us to select the proper devclopement for ^r. Thus, if a 7 I 



