88 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XIV. 



Table XVc — Continued. 



Unit=l" 



An inspection of the preceding table, which is t}"pical of all the trigonometric series under 

 consideration, shows readily that any function of this type is of the form 



II-' sin K' + Iksin {K±iP)==IJc' sin K' + lie sin K cos <!; + II' cos K sin (J) 

 or 



II-' cos K' + Ilc cos iE±tl>)=Il-' cos K' + Ilc cos 5: cos </>+ II- sin Ksin ^ 



or, more briefly, a + & cos ^ + c sin 4> 



wherea, 6, c are trigonometric series and can be ^vl■itten by inspection from the tabulated function. 

 Hence, in v. Zeipel's notation (Z 54, eq. 96), 



Ti = Xf+Yi cos 4> + Zi sin (p 

 and the integral may be written 



Wi==Xi+yi cos ^ + Zi sin ([> 



The functions T and W are to be used in this form in solving equations (95). 

 Considering only first order in the mass in T 



T, = X^ + Y^ cos ^ + Z2 sin ^ 

 X.^^IJc' sin K'; Y^ = I1c sin E; Z,= ±I]c cos E 

 or, Xj is the part of Tj which is independent of ^, Y^ is a trigonometric sine series having the 

 same numerical coefHcients as the part of T^ which contains (p in the argument, but in which 

 (p is omitted from the argtmaent, and Z, is the corresponding cosine series. 



Considering the first two of the eqs. (95), the first one states that IFjisnot a function of £ 



a^o"''' 01"' W, - [ FJ = ; If, = [ FJ. 



Making use of this fact in the second, F, can be obtained from (dB^). (See Z 54.) Introducing 

 the auxiliary functions ^-j and w,, defined by (99) and (101), the difTerential equation for F, is 

 replaced by the ec^uivalent differential equations, (100) and (102), for ^, and u^. 

 The series r yj — 71 [ YJ 



a"^' [Y,]cos,p + [ZJsin4> 



can be written by inspection from T^, or, better, the integration itself can be performed in part 

 at the same time. 



where 



