92 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv 



After the determination of Wj, the function W, — [TF2] is obtained from the solution of 

 Z 53, eq. (95j). The integral may be written as in Z 63, eqs. (10.5), (106), or, quite as simply, 

 as follows: 



F, - [ Fj = F/ -f r ( r, - [ r,])c7£ 



K'= - rj|(l-fi cos b){w+ TT;)-i[(l-c COS e){w+ Fj]W> de 



The function F' is given in Table XIX. 



Anticipating some later developments, for which we shall need 



[(l-gcosc)lF! 

 the function _^ 



[(l-ecos£)ir/] 

 is tabulated in Table XX. 



The determination of [ IF,] °^ay be accomplished according to Z 65, eq. (108) —Z 67, eq. (116),. 

 or in the manner outhned below, which we regard as preferable. 



Repeating Z 65, eq. (107), 



<^/^^ + fe-W. + W4j = j^<^.+|[(T^t-| E,)(W,+|H,)(l-e cos^)] 



— (1 — e cos e)\{w 



^+W^)^(,jiT,-[T,])de + ^j(T,-[TJ)dJ^ + 2[T,] 



in which all the known parts are contained on the right-hand side, the development of equivalent 

 equations proceeds in a manner analogous to that for W^. 

 Writing 



T3 = X3 + Y3 cos (/i + Z3 sin (p 



and introducing 



<^2 = [^2]-'?[y=]+w' 



and equating parts independent of tp, coefficients of cos ^ and coefficients of sin (J), the three- 

 equivalent equations arc: 



d[x.] dx^ ^ fZ.r, , sr,, V ™ 1 „ V w , 1 - M'^^ii 



-[(1 -e cos e){w+ W\)^gj(X,-[X^)de'j-^il-e cos e) f (T,-[T,])d^ll^ + 2lX,] 



^d[y.] , ^dv, ^(^Vi^^Vf, Yir 1 - V fif ^ 1 « M^-Vi 



*'^+'^^rf^ = ^"^'W + 4L^^-^'^°^^\^'-3-A^^' + 9"0J^ 



-[(1-e cos e){w+ W\)^J{Y,-[Y,])de'j-^{l-e cos e)J {T,-[T,])dsJ^ +2[Y^ 



-[(1-e cos e) {w+ ]]\)^,j{Z,-[Z,])de'j-^{l-e cos e) j {T,~[T.^)de'^ + -2[Z,]^ 

 Multiplying the second of these by tj and subtracting from the lust: 



-[(1 -c cos o(w+ iTo^|J{A',-,r,-[X3-,rj}j£] 

 -[(i-e cos £)J(7;-[7;])(?£]'^+L-[A'3-,r3]- 



