596 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



or sin X , cos f — + f ) — cos ( ~ !. + •'^ ) 



- ( COS — =fc:V - 1 sm -^ 1 s"i -^ ( COS -:^ ± V - 1 sin — j^— j cos I -g- + "^ ) ' 



( cos -^ =t V — 1 sin -J- ) 5 cos ( — -rr + ••■ ) 



The number of values which r admits of is the same as the denominator of 

 the fraction which stands as the index of differentiation. 



Hence, \S.p be the denominator of the index, both r and r' admit of j9 different 

 values ; and any one value of r may be combined with any one value of r', so that 

 the number of different differential coefficients is p-. 



This result differs from that previously given, M. Liouville having adopted 

 a very indirect process for obtaining the different values. 



24. To find the differential coefficients of the other circular functions of x. 



We have no other process than that of expansion, which, of course, will not 



give a complete result. Thus, to find the wth differential coefficient of tan x we 



may proceed as follows : 



rf" tan X d" sin x ii d d""^ sin x 



— 3—- — — sec X — -7— H —. — sec x . j— — r 1- . . . 



rfa;" dx^ dx dz'^'- 



the results of the differentiations being supplied by the above formulae. 



The values of the differential coefficients of the inverse functions must be 

 determined by a similar process, but it will not be necessary to write them down. 



We shall, however, give the differential coefficient of tan ~' x, as it may be 

 done in a very simple manner, when n is positive and greater than 1. 



Let M = tan ~' X. 



du _ 1 _ 1 1 



1 \ 



2 Vl + ^/ITT.. 



l-V-l.a; 



d" u «?""' du 



dxT daf-^ dz 

 1 rf'-i r 1 



■ + 



7} 



2 dx"-^ \ y 

 if y~\ + s/ — \.x y'-l-V -1. X 



Now, (/y'-i = (V"^)"-i(/^"-\ {dy')"-^ = {-\/^y'-^dx"-^ 



d"u 1 , . fl!"-i 1 1 . (^-1 1 



rf«" 2 ^"^ '••^ ■ rfy-i ■ y ' '2^ ^ ~'> ■ rfy"-i 



=4-(V-l)"-' . (-1)"-' .-^ + l(-^/-l)"-V-lr-4!r 

 ^ y -i y 



