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By I. J. Schwatt

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1. tj(*) ( |; [J"" where ' ( parts, (113) ^ - ( 113 > (114) becomes (us) _ o~ v (" 1 >\%0 ) (- 1 P ) cot20a; &+l) cot2ma; ( - 116 > (117) . DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 39 Therefore d 2n 2» -\) k /2n + l\ /k\ ^cosec = (-l)"gl^(^ /)2Q(i-2a^cosec»-«« ( k a: i>f2 ,, (118) ^(^(^-Sa^+icosec^+^i^/s+i. ] (119), we obtain ( l -^(i:i)| 0 o^^— s <-^; 0=0 where (ii) y Another form yx = I. >» where and, by Ch. ^ = l-(-l)' for the higher derivative of y Now cot2fl+ 7) e 2ix _ : < 121 > (62) and i (83), Hence Now, applying to (124) the method by which we have from (121), (64) was obtained from (63), fin,, n pZiax a /_\ ^=^^(^^^(-1)^(3(1+2^.

If y o s („ ! k) ftf) - ^((^x + 1 *) <-> w ?

119), we obtain ( l -^(i:i)| 0 o^^— s <-^; 0=0 where (ii) y Another form yx = I. >» where and, by Ch. ^ = l-(-l)' for the higher derivative of y Now cot2fl+ 7) e 2ix _ : < 121 > (62) and i (83), Hence Now, applying to (124) the method by which we have from (121), (64) was obtained from (63), fin,, n pZiax a /_\ ^=^^(^^^(-1)^(3(1+2^. But and (^nji-^C 1 -^^)" (1 -icota:) a = Y -^2 +i, il/2 Y (125) ( 126 > (127) where and i^+i are the expressions in (93) and (94) respectively, except that a takes the place of lc - 1.

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