540 Mr. J. W. L. Glaisher on some Identities 



bers, and, after the first, the terms are 'alternately negative and 

 positive in pairs. 



Treating other formulas in the Fundamenta Nova in the same 

 manner, it will be found that we have 



arc tan q — arc tan q 3 + arc tan q h — arc tan q 7 + &c. 



- arc tan t g - g3 ~ gl5 + g21 + g45 - &C - 

 -arc t an ol _^_^ + ^ 28 + ^ 36 _ &c> 



- i arc tan 2g + 2g9 + 2 g 25 + 2 ^ ' 9 + &C * 

 ~ 2 l+2g 4 + 29 16 + 2g s6 ~T&c7 



- tare tan 3? + % 3 -l V 5 -^-{-&c. 

 ~~ 3 l + 7q 6 -9q l0 -l5q™ + l7q 36 +&c^ 

 Jl V_ + _i2!__ &c 



= \ arc tan 



1 i +9 4 + i+? 8 i + ? 18 



In the third result the exponents are the triangular numbers, 

 the signs are positive and negative in pairs,' and the coefficients 

 are the uneven numbers. 



Denoting arc tan q — arc tan q 3 + arc tan <? 5 — &c. by i/r, we can 

 easily find also expressions for tan 6^, tan Syjr } tan 12i/r, and 

 tan 16^; for if P* denote what V becomes when qi is therein 

 written for T, we have 



, -, , ! r /2Kn 2 /2K\ 3 /2K\ (?Wk.K\ 3 

 and the formulas for ( — 1 , ( — J , I J , I — J are given 



in the Fundamenta Nova, pp. 103, 108, and 115 ; but the ex- 

 pressions are not of sufficient interest to make it worth while 

 to write them down. I may, however, add the formula ob- 

 tained for tan 4^/r by means of the value of f c ' \ ,for 



comparison with that given above, which is derived from the for- 



2K 



mula for . The numerator of tan fair is found to be 



7T 



qs/q 3\/q s 5q b \/ q b __ 7 */ q 1 



