wo MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 



and denote first order terms in TF3 — [W3] and W^ — [TFJ by W3" and W^", respectively. 

 Then because of the similarity in the equations for these functions of successive ranks, the 

 sum 



can be computed by Z 70, eqs. (117), (118), (119). The coefficients F, G, B are tabulated in 

 Tables XXIII, XXIV, XXV. The mass factor m' is, of course, implicitly contained in the 

 tables. 



Ehminating the distinction between (p and s, the function is 



W,"+W,"+W," 



in which the coefficients Apg, determined by Z 71, eq. (121), are tabulated in Table XXVI. 

 The coefficients A^g in the function 



(1-f cos«) {W,"+W,"+W/') 



are computed by Z 71, eq. (123) and are tabulated in Table XXVIL 

 By means of Table XXVII we readily compute 



[(l-fcos£) iW," + W," + W,")] 

 tabulated in Table XX\1II. 



Proceeding now to the determination of 



[(l-<'C0S£)ll^] 



(from which we shall subtract [(1 — e cos) W^"], already included in Table XXV ill), we have 

 by Z 53, eq. (95) 



^^W,-[W,])^{T,-m) il-e cos e)iw+W,)^'-l'^^'[i^-e cos s){w,-w-W^) 

 -|(l-eco3e)(W.-iSi) (W. + i^.)- [d-e coss){w+ F,)^'] 

 -[(l-ecos£)(F,-wW;)]-|[(l-ecos£)(F,-i5,)(F. + |^.)]} 



in which all quantities are known. The integration gives W^ — [ IFJ. 



Having computed W^ - [ TF3], [ Wj,] can be obtained from Z 53, eq. (95). 



[(1 -£ cos £)(w+ F,)]^^V[(1 -e cos £)[ W3]]^' = 2[r,]-[(1 -e cos £)( F,-wF,)^'] 



+ [1 - e cos e)w W,t^ -[{1-e cos e) (w + W,)^g{W, - [ W,])] - [(1 - e cos .) (TT^- [iF,])]^' 



The function [TJ, computed from Z 53, eq. (94), is tabulated in Table XXVIIIff. 



In a manner similar to the development of equations for W, and f W^], the right-hand side 

 of this equation, when computed, can be segregated into portions independent of 4>, terms 

 multiplied by cos <l>, and terms multiplied by sin (p. It is of the form 



A + B cos ip+Csin (J) 



where A, B, C are too complicated to be written analytically, but can be written by inspection 

 after the computation has been performed. 



The equation can then be written in the three following equivalent equations: 



^.%3+(^3— ')^ = ^ 



in which we define 



