of certain Branches of Analysis. 34.9 



i cot — = sin X + sin 2x + sin 3x &c. 

 ]og(l +ry,x) = _ log2 + I C05X - I C05 2X + I co. 3x &c. 

 - = Sin. X ^_ + JL"^3JL _ &c. 



"^ — ^ ■ sin 1 X sill T V 



-^ = Sin X + -^_ + _i^l 4. g,^,_ 



By thio simple means, a vast number of different processes 

 may be reduced to one transformation; and were proof wanting 

 ot the smipl.city as well as unity which an attention to it be- 

 stows on many investigations. I would compare the above 

 with the mvestigations of the series for log(l + cos^) given 

 bv Euler and Lacroix in their great works on the Intl-ral 

 Calculus. » 



But all this is easy and comparatively ti-ifling. The r^reat 

 problem before us requires the development of any function 

 of z, i}^ into a series whose terms are of the form (3). This 

 may evidently be accomplished by decomposing B into the 

 sum or difference of two similar functions, one of 2 and the 

 other of 2 ', or by solving the functional equation 

 f s ± ^z~^ = B, 



I will not enter into the general solution in this paper, but 

 will prej)are the way tor several methods of development which 

 1 mean to explain afterwards by the full consideration of a 

 j)articular case. Let us investigate the general form of A 2 

 when B^=0; 



that is, let us determine the two classes of functions which en- 

 joy the properties 



^z = (pz~^ 

 (pz = — cpz~^ 

 The first of these equations simply indicates that <t is not 

 changed by the substitution of 2"' fo,- z- and since -"^ and - " 

 are convertible into each other by mutual substitution,' it i^ 

 lilam that this ciiaractenznig condition will be fulfilled by sup- 

 posing ^ a sipnmetrical function of z and 2~^ It is pvul^nt 

 U can be fulfilled in no other manner. Hence if / re,T" sen 

 any arbitrary form, we have for a ireneral solution ' 



Agam, f z ni the second equation must for similar reasons be 

 a symmetrical function of. and z-\ multiplied by some other 

 arrangement of them which changes its sign when (I,c quan" 



tilics 



