Exponential and Logarithmic Theorems. 45 



•1 1 — q— ^ ) X&c, which must bean identical equation, its second 

 member being the factors of the first member; or omitting the 



factor V, and putting —=z^ we get 1 - Y9r\^'^ i "2 ^ a ^^^ ~^^' 



= (1 — z). I 1 — ^ ) ( 1 --5- j X&c. ; .'. by equating the co-effi- 

 cients of the same powers of z., in the two members of the equa- 

 tion, using the notation in (a'), and the results in (e), we get 



7r2 / 1 1 1 \ TT* 7T6 



A = -^ = -^l+j+9 + i^+&c.j,B = j^,C=-^^^,andso 



111 n^ 111 



on; .-. l+2^+3-2+4^+&c.=-^, a=l+^+37+j7 + &c.= 



A=-2B=^, -R = l+,^+3^+&c.=:-(A3-3AB + 3C)=^, 



and so on ; which agree with the results obtained by Euler at 

 p. 131 of his Analysis of Infinites: and by comparing the value 

 of cos.v given in (A) with its value given by (B^), we have 



11 n^ 1 1 tt" 



l+3^+5^+&c.=-g-, l + 3^+^+&c.=Q^, andsoon; as Eu- 

 ler has given them at p. 132. We shall now resume M=(l+1). 

 ( l-fn)-( 1+5) X . . . (l + iyi _ A : and by supposing M to be in- 

 finite, we have M=(l+l).(l+^j -(1+3) -[l+^j X &c. to in- 

 finity; .'. by taking the hyperbolic logarithm, we get L.M = 



/111 \ If 1 1 ] 



\l+2+3+4 + &c. to infj-2|l+2^+37 4-&c. to inf -f 



3 1+2^+3-3 +&C. tomf -&c., or L.M=^^l^-2 + 3+&c.j- 

 ^l!' 1 1 1 If 1 1 ^ 1 

 [_2 l+2l+3i-+&c. -3 l+2^+37 + &c. -f&c.J, where the 



quantity within the braces is evidently positive, we shall denote 

 it by A, and we shall have I4-5 + 3 + T+&C. toinfin.=L.M-|-A, 



.'. since M is infinite its logarithm is infinite, .'. the sum of the 



11 



series 14-q-1-o+&'C. continued to infinity is infinite. 



