ANALYTICAL FORKVLJE. 135 



1*. 3*. 5*... to infinity 



nity r 2^--/3' \ C^Jl\ ^ 



= A X 2*. 4*. 6*... to infii 



( ^ ^ X &'^« Now from this we djerive, by taking as before 



the logarithmic differentials on both sides 



d A tan. A sin. 2 A 



"A "^ dA 



(5*—^-^)^ .of 1 . 2 A 



* ■ (SO 7'^^;,w 



A 1 



If in this form for A we write -r-, and divide by 2, it becomes -. r- 



which is the third formula I designed to demonstrate. 



If in [}) for «« we write a, and transpose &c., we obtain 



-i— + —L. + -^ + &c. = C-i — cot. A^ X -^;or, 

 1>— rt ^ 2«-a ^ 3«-^a ^ VA ^ '^ 2 A ' 



ffriting for A, ^ Va, ^ + ^i;:;^ + ^j^;;^ + &c. == ^ - 



^X cot.^v'a 1X1 



In (2) for ll* write a, and for A, its value — v'"a ; »t becomes 



-i— + -^ + -^— + &c. =: -4= X tan, — V o ITl 



By multiplying: 



