362 



REPORT — 1865. 



formation of a form (a, b, c) is necessarily one of the five following : — 



therefore, conversely, a reduced form can be transformed into a form («, b, c) 

 only by one or more of the transformations, 



1,0 1, +1 0,-1 



0, 1 ' 0, 1 ' 1, +1 



and upon trial it will be found that there is always one, and only one among 

 them which applied to a reduced form, produces a form (a, b, c). The number 

 of forms (a, b, c) is therefore equal to the number of reduced forms of deter- 

 minant — (4n + 3); i.e. F'(4u + 3)=F(4/i+3). Eliminating J, we obtain 

 the first and third formulae of M. Hermite's memoir, 



OO 4n+3 GO 



Vl ZF(4n + 3)q { =22[A(»)-rx(»)]2", 

 1 



00 4«+3 00 4n+3 



02F(4n + 3)<z 4 =|S(--l) n [*(4»+3)^¥(4»+3)]2 4 , 







or, equating coefficients, 



F(4n-l 2 )+F(4/i-3 2 ) + F(4ji-5 2 ) + . . . 



= A(«)-r 1 (», 

 F(4»i + 3)-2F(4>i+3-2 2 )4-2F(4n4-3-4 2 ) — ... 



= (-l)"|[*(4«4-3) -¥ (4n+3)]. 

 In his second memoir M. Hermite occupies himself with the demonstration 

 of the equation (B'). Multiplying together the two series, 



GO 



„., eH, , w -i v. +/' sin 2,uv 



CO 



fl 1 l®!=tana; + 2S (-ly^Q^^'^sin 2nx, 

 H x 1 



and employing the formulae 



he finds 



f 



r 



sin 2nx cot x dx=ir, 



sin 2nx tan xdx=( — 1) ir, 



el=l 61 redx=l ffl! |5* X 0, %S dx 



* Jo "Jo 6lK Hl 



co „ co 



= 1 + 4 f lT( !:i).- 5 .- 2 f- 1 )- Q -.s- : 



T ll+(-l)Y' 

 an expression which, by a detailed discussion, is shown to be equivalent to 



co 

 122 E(«)2 rt . 







