TRANSACTIONS OF SECTION A. 271 



As a particular case, put a = l, 6 = 2; and this equation gives 



.j.2n .j.2n ^.2n 



arc tan V- + arc tan — - + arc tan — . + &c. 



^2n y2n <j2m 



sin (w.rA,) sinli (7r.rB s ) 



«=o cos (7raA s ) cosh (ttxB $ ) + 1 



= i 2 s=n ~ ! . sin (w-yA,) sinh (ttxB,) 

 s=o cos (jr.rA g ) cosh (ttxB s ) + 1. 

 It can also be shown by the method employed above that 



£.2n 3-2n .p2n 



arc tan ,=— + arc tan — - + arc tan '— - + &c. 



]2n 32n g2n 



= 2 Lo~ arctan I tan (* ' r;rA J tanl1 (i»r^B,) } 

 which is readily connected with the result just written ; and that 



-p2n yln r^ln 



arc tan —- + arc tan ' + arc tan — - + &c. 

 \2n 2 2n 6 2n 



= ( _ y-i j „ _ s « =B - 1 aic tan tanh ( ff ^ j 

 s=0 tan (n-.rAJ 



Of course the two sides of these equations may differ by any multiple of w. 

 As particular cases, by putting n = 1 and 2, we have 



arctan L. + arc tan — + arc tan — + &c. = arc tan J tan tanh irx I 



F 3 2 5 2 \ 2^2 2^2 j 



x 2 x 2 v 2 tanh — — 



arc tan — + arc tan — + arc tan _ + &c. = £ n - arc tan \/f 



1 ™ .. 



tan — 



•7^4 o^4 ~A 



arctan — + arc tan — + arc tan — + &c, 

 l 4 3 4 5 4 



= arc tan 



tan (^ ttxo) tanh (£ 7r.r/3) [ — arc tan . 



tan (| Tr.r/3) tanh Q ttxo) 



x^ x^ x^ 



arctan -. + arctan — + arc tan ~ + &c. 

 I 4 2 4 3 4 



= - i tt + arc tan tanh ^ Ta ) - arc tan tan]l (^ 

 tan (7r.r/3 tan (jrxa) 



where a = cos £tt, /3 = sin £n- 



formulae which are given in the Quarterly Journal of Mathematics, vol. xv., pp. 

 151-157. The general formulae to which this note refers are more fully discussed 

 in a paper which will appear in the Quarterly Journal, vol. xvi. 1 



12. Note on a Method of checking Calculations. By W. H. Walenn. 3 



The object of this note is to draw attention to the advantages of checking cal- 

 culations by casting out the elevens in preference to casting out the nine's. 



By this means, in many cases, the calculation can be checked by an appeal to 

 the question and answer only. Also decimal calculations can be checked in 



1 Addition to a paper 'A Theorem in Trigonometry,' vol. xvi. (No. 64.) 



