200 Prof. Young on the Development of certain 



-t—-, and therefore the following coefficients, are negative ; 



when it terminates in the fourth quadrant they are positive, as 

 at first; when in the fifth negative, and so on. Hence the 

 general expression for the development of sin -1 x, terminating 

 in the k+ lth quadrant, is 



r ifi 3 g ,r 5 3 9 5 9 * 7 ; x \ 



**±{*+ I^ + l^X5 + 1.2.3.4.5.6.7 + &C 'i 

 the upper sign being used when k+l is odd, and the lower 

 when it is even, £+1 being either positive or negative. 



The series for y = cos"" 1 ^ is inferred from the following 

 conditions, which are said to have place when x = 0, viz. 



T-z ^---i ^/-o^-i&c 



sin -1 .27 



2' rf* "" ' do?* ~ ' d 



But the first of these conditions is true only when the pro- 

 posed arc, 3/, terminates in the first quadrant. If it termi- 



3fl" 

 nate in the fourth, y must be ^r— , when x = ; if it termi- 



m 



nate in the fifth, y must be % —— ; if in the eighth, y must be 



sr* and so on. Moreover, in the fourth quadrant -^-. —4, 

 2 *■ ax ax* 



&c., have signs contrary to those which they have in the first ; 

 in the fifth the signs are the same as in the first; in the eighth 

 opposite, and so on. Hence the general expression for the 

 development of cos- 1 x, is 



cos * .r= hr 





the arc terminating in the &+lth quadrant; the upper sign 

 being used when k+l is odd, and the lower when it is even. 

 It is obvious that if the arc be negative this expression will 

 take the minus sign. 



By attending to similar considerations we shall find, for the 

 development of tan -1 .r, the general expression 



17 X* X 5 X 7 Q 



tan" 1 .r = fcTT + x — - + — — -f, &c, 



o o 7 



the arc always terminating in the 2 £4- 1th quadrant. 



The development of tan"" 1 ^ is frequently deduced by the 

 aid of a certain logarithmic formula, without the application 

 of the calculus; the resulting form is, however, limited to the 

 single case above mentioned ; and this limitation arises, as in 

 the former process, from an oversight in the investigation, al- 

 though one of a very different kind. The investigation we 



