350 Outline of general Methods for the Development 



titles are transposed. Whence an equally general solution 

 for the second case 



Bv means of these two expressions we may obtain as many 

 series as we please of the sines and cosines of multiple arcs 

 whose values are 0. But before illustrating the method by any 

 particular case, let us see if we can connect with each case any 

 class of series of a similar description. It is evident that if 

 we could derive a series of functions of the character 



^„.±^^z-^=D(?_,.±^_,::-^) (6) 



where (p^z ± ?!,^3~^ = D"(^?o= ± fo ~~' ) 



we should have a new and extensive orderof trigonometricfuuc- 

 tions hanging upon each form assumed by D and (p^ or (p. Not 

 to complicate the matter further than is necessary for the pur- 

 poses of illustration, let us give D a particular form, and deter- 

 mine D"' in series by a process which, although particular, 

 contains the elements of treatment that will be applicable to 

 every case. The values of .p s + ip ~"~ ' as characterized in 

 equations (4) and (5) are favourable to the determination of D" 

 when D is an integral. Let the formula of derivation from 

 <p z in the first instance, as characterized by equation (4), be 



therefore <P,i^ = fi' ^ f^~ <Pn-i^ C^) 



arising from <PiZ = f -^t z d z (p z 



And let us modify 4/ z rf s so that the condition (6) may be ful- 

 filled by it. Substituting z~ for z in equation (7) we have 



^^z=f^z-'dz-'<p,^_,z-^ 

 whence 



<p^Z ± f^z''^ =/'(^-^'^?„_l^ ± vl'-~''^Z~' ^n-l^~^) 



where it is evident the requisite condition will be fulfilled if 



±'^=^z (8) 



Supposing then that vj/ 2 is thus characterized, the formula of 

 derivation becomes 



noting by u c the constant arising from integration. From 

 this formula we immediately derive by successive substitution 



