486 SCIENCE PROGRESS 



_.„ dbasx , , dbassx . , 

 13. -^- = £bas^ __ = _ 4b as:r 



dbas'^ 4 



dx s/4-x* 



AAA A A 



1-4. bas^=basj- .basj— .basj-^- .... basj— .bas— 



245 2" 2" 



= bas^ . bas^±^ . bas 4 ^±A . ba s^±^ . . b J n ~ l)27r + A 

 n n n n ' n 



basA . bas 2A . bas 3A ... . bas«y4=bas(«+ i)A. 

 1*5. basjr = 2 bas^n- = o 



bas^ = sj2 bas^ = -Ji- J 2 bas| = V2- JT^f 2 



bas^ = i bas^=\/2- ^3 bas— = ^2- v/ 2 + ^3 



as '5 = ^T -1 bas^=v/ 2 -| v / IO+2 ^ 



r6. Prove that bas0<0 and bas 6>\6 . bas s0 ; or bas 2 0> -^. So that 



4 + p 



0>basfl> 2 * . Put0=-. Then 

 V4 + 6 2 n 



\>\>*s-> , *" . Difference is ? - 7 27r =<r /l \ 



V2«/ 



7. (bas.r<9 + *bas0) n = bas.y;z0 + *bas/z<9 by De Moivre's theorem, 

 •o. For the solution of triangles, if a, b, c are the sides and A, B, C the opposite 

 angles, we have 



a _ bas 2 A __ bas A bas sA 



b bas 2B ~ bas B bas sB 



a 2 = b t + e i ~be(2-bas i A) = (b-e) 2 + bebas^A = b^ + e 2 -be^ S ^4- 



bas 2^4 



It is useful to write bas(£, c, A) = y(b-e) 2 + bebas 2 (i, 1, A). Then 



, ,, A , , bas 2/4 bas 2^4 



bas( *' '' ^ = *ba72i? = C bT^C> etC - 



Also basM = 4 ( '-^~ C) bas^ = 4 '-^- ) 



^ be H be 



where s is half the sum of the sides. 

 3 - o. Exponential values. 



i bas 2x = e ix -e~ ix bas S2x = e ix + e- ix 



bash 2x = e x -e~ x bash S2x = e x + e~ x . 



4*0. Iteration (see Science Progress, October 191 8). 



[basO]^0-^ + ^^g;- ^ 75 " 2 - 33 ^ +l64 ^ + etc. 



L J 2! 2 2 5! 2 4 37! 2 a 



basl —J =bas— = [\/2- ^-o^basO (from r2). 



bas nir . v/4 - O 2 4- bas snir . O = [ v/4 - O 2 ]". 



[Note. — It will be observed that the new functions have a periodicity of 477 ; 

 and hence, to a value of bas x (say) between o and 71-, only one value of x belongs, 

 whereas to sin x (say) two values belong. Thus it seems that these interesting 

 functions may be also of practical importance in avoiding " the ambiguous case " 

 in the solution of triangles. — P. E. B. J.] 



