REPORT ON CERTAIN BRANCHES OF ANALYSIS. 281 



J -^ p 2 s/ah ^ \dx ^ / 



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

 = — 1 and X = + 1, we shall find, 



r + 'dx_ 1 , ii 2r i/^-V 4a6 - (« + 6) (1 + ah) '} 

 ^"J-i p ~ ^^ab^^t2r' xrab-i>ab-{a + b){\+ab)y 



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



a X ^ 



— 4 a 6 — (a + 6) (1 -\- ab)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 + b\ 

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

 must have the same sign with (1 + «)(! + b), and consequently 

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



g ^ _JL^ log . {a - 1) {b - 1) '/ a b + 4 « 6 - (a 4- 6) (1 + a6) 

 4^a6 (a + l)(6 + l) v'"«6-4a6-(o + i)(l +a6) 



= -. — r=T log . \ — = L (striking out the common divisor 



4>/a6 "= {s/ab-lf ^ ^ 



-_ 1 , ^/a~b + 1 



2 Vab — a-b) = ^ , , log —i=r 7 = ± ^4- 



' 2 \/a b -/« o — 1 



If a i^ 1 and 6 ^ 1, we shall find r = (1 — a) (1 — 6), and 



1 , { \ + VTb \ , 



z = = log I = ! = +«,. 



2^/ab ^\i- Vab^ ~ 



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



z = 7^= log I —77 -r- ) = + Zq. 



2 Vab *= \Vb- VaJ " ^ 



If a 7\ and b /il, we shall find r = (a — 1) (1 — b), and 



1 , /\/« + Vb\ , 



It would thus appear that the definite integral would furnish 

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

 of the factors of r and r', 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 z. Thus, if a 7 1 



