of certain Branches of Analysis. 353 



will satisfy all the conditions which restrict it, if we suppose it 

 equal to 



^^= '"7^+ ^-^ + &c. (II) 



It might be seen from a foregoing part of the paper that a, 1, 



l^--y +-^- ^+&c. 

 is equivalent to -^ when radius l....The right hand member 

 of equation (9) is thus referred to circular arcs, and loga- 

 rithms of z. Let us now separate the cases in which n is 

 an even or odd number, and we have from (10) and (11) 



u (c + *) = 



u (c + *) = 



1m\\ 



which if z = f'"'^ ' give us the series 



±. »J_c +*) = cos x-^f + ^-- ace. 



It is now time to conclude. It must not be conceived that 

 all this analytical labour has been spent merely in the investi- 

 gation of the foregoing two series ; — interesting as they are, 

 they might in this case be thought dearly obtained. I have at- 

 tempted throughout merely to sketch the leading features of 

 all similar investigations, and to leave untouched only such 

 (lifTlcidties as an analyst of very moderate pretensions may 

 grap|)le with. The field which I have sought to point out, 

 lias tu be explored ; and 1 am convinced that the contents of 



Vol. io5. No. '6'15. May 1 825. Y y it 



