122 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv. 



If we observe the character of d as it is expressed in the definition and recall that we have 

 admitted trigonometric terms in 6, mxdtiplied by t, it is evident that this argument, which is 

 a function of the disturbed positions of the planet and Jupiter, is not periodic, but varies con- 

 tinuously with the time. In the foregoing equation g and g' can not be regarded as angles 

 which are always less than 3(30°. contains, therefore, a nontrigonometric secular part in s 

 and a periodic part in and e. 



If we write 



d — [d] contams the secular term in c as well as periodic terms. The segregation of terms of 

 different type can be made explicit by the introduction of 



d = ^ + 0,{^,s)+d^{&,i)+d^{d,s)+ Z78, eq. (125) 



where t? is a function of £ and 5„ 62, 6^ • ■ ■ are the periodic parts of 6 — [6], i. e., they are 

 entirely trigonometric functions of s. This covers the condition that Oi can not include trigo- 

 nometric secular terms in s. By definition of § and (?< 



y 



^' = {F(0,e) - [F{0,e)]) - j^(n'8z' - [n'Sz']) 



where [n'dz'] is the long period term between Jupiter and Saturn, 

 The derivative of (125) is 



dd^dd dddd_ 

 de hi d^ds 



/d6 be dd, VA ,^^.,^^^4.^^3, Y& 



~VdI+d£+d7+ ; + V d^+d^+d^ + Jde 



Expanding F{d,s), eq. (124) in a Taylor's series in ascending powers of ^<and making the above 



substitution iov-j^' (124) becomes (126), in which 



F{d,^) = \{l-eco5i)[w+{\-w)W-\{\-ui)(W-\s^(^W+\E^+----^; 6 = ^ 



From the Taylor's series -^ is written m (127). This is the differential equation for 1?, the 



right-hand side of which can be computed. 



Substituting ;5^ in (126) and equating fimctions of equal rank, we have the differential 



equations (128,-1283) for di, which can be integrated in succession. 



Before integration we convert eqs. (128) into differential equations for ndz as follows: 



Let 



nhz = (ndz — [ndz]) + [ndz] 



= ndzi+ ndz^ + ndz^-] [-[ndz] Z 88, eq. (144), 



where ndzt is not only a function of fiist and higher orders in m', in which the lowest rank is i, 

 but is entirely trigonometric or periodic. Then 



Z 9, eq.(46) gives n5z-[nSz] = j^f-;^|(?,(j?,s) +<?2(i?,£) +^3(i?,£) + +wrj sin s+ (n'dz' -[n'dz'])j 



and 



M2] = y^{.>-|e + [7j,'a2'] + <-'-A(c} Z 88, eq. (145), 



where it is to be noticed that [ndz], unlike [ TF], is not free from terms in s. Subdividing the first 

 of these two equations according to rank, we have Z 79, eqs. (130), in which — n'dz' + [n'dz'] can 

 be neglected. 



