THE ORBIT OP NEPTUNE. 



33 



Sp = — l".llsm{2r — l — n+r) 



— .72 sin (2 Z' — Z — 7t — t) 

 + .16 sin (2 I' — I — n'-\- r) 

 + 0.15sin(2Z' — Z — 7t'— t) 



— 2".98"sin(4Z' — 2Z — t) 



+ 0".0110 1 



8qzz — l".ll cos (2 Z' — Z — 7t + t) 

 + 0.72 cos (2Z' — Z — 7t — t) 

 + 0.16 cos (2Z' — Z — ti'+t) 



— .15 cos (2 Z' — Z — 71'— t) 



— 2".98cos(4Z' — 2Z — t) 



+ 0".0001Z 



§ 16. Perturbations of the co-ordi7iates — Comparison with Peirce and Kottalski. 

 — The first column of the following tables gives the coefficients according to 

 Peirce (Proceedings of the American Academy, Vol. 1, pp. 287-291) ; and the 

 second, the values according to Kowalski (Recherches sur les mouvements de 

 Neptune, pp. 14-16). In the case of Uranus, Peirce's coefficients have been 

 increased by | + -gJ^, to reduce his mass of Uranus to the adopted one. The 

 coefficients enclosed in parentheses are not comparable, as they include the effect, 

 of terms now developed as perturbations of the elements, and therefore omitted 

 from the perturbations of the co-ordinates. The perturbations of the radius 

 vector have been reduced to logarithms by multiplying by '^gV-^. 



I.— ACTION OF URANUS. 



^logr = 



P. 





K. 



N. 





p. 





K. 



N. 









(— 206".91) (_244".40) + 3".002 sin {V — I) 



{- 



2284) 



(- 



2289) 



+ 314 cos 



(Z'- 







+ 10 .24 



+ 



10 .02 



+ 9 .994sin2(Z' — 



+ 



167 



+ 



163 



+ 162 cos 



2(Z'- 



-Z) 





+ 2 .01 



+ 



2 .02 



.+ 1 .960sin3(Z' — Z) 



+ 



40 



+ 



09 



+ 38 cos 



3(Z'- 







+ .64 



+ 



.62 



+ .610 sin 4 {V — I) 



+ 



14 



+ 



88 



+ 13 cos 



4(Z'- 







+ .25 



+ 



.27 



+ .237sin5(Z' — Z) ' 



+ 



5 



+ 



23 



+ 5 cos 



5(Z'- 







+ .11 



+ 



35 



+ .104sin6(Z' — Z) 



+ 



2 



+ 



U 



+ 1 cos 



6(Z'- 







+ .05 



+ 



.27 



+ .041 sin 7 (Z' — Z) 



















+ .02 







+ .017 sin 8 \v — I) " 



















+ .01 







+ .007 sin 9 \v — V) 



+ 0".002 sin (— 4 Z' + 4 Z — tt" + 

 + .016 sin (— 3 Z' + 3 Z — tt' + 



tt) 

















(- 0.11) 



(- 



0.73) 



— .103 sin (— 2 Z' + 2 Z — tt' + 



tt) 

















(—16.29) 



(- 



16.79) 



— .048 sin (— V -\- Z — V + 



tt) 

















(+ 0.66) 



( + 



0.71) 



+ .045 sin ( V — Z — tt' + 

 — 0.011sin( 2Z' — 2Z — 71^ + 

 + .003 sin ( 3 Z' — 3 Z — tt' + 

 + 0.003sii( 4Z'-f4Z — V + 

 + .002 sin ( 5 Z' + 5 Z — tt' + 



tt) 

 tt) 



tt) 



- 















— 0.01 







— 0".009 sin (— 5 Z' + 6 Z — tt) 



















— 0.01 







— .014 sin (— 4 Z' + 5 Z — jt) 



















— 0.02 







— .024 sin (— 3 Z' + 4 Z — tt) 



















— 0.04 





-0.08 



— .033 sin (— 2 Z' + 3 Z — tt) 



















+ 0.19 



+ 0.19 



+ 0.183 sin (— Z' + 2Z — tt) 



+ 



2 



+ 



2 



+ 3 cos 



[— '' 



+ 2Z- 



TT) 



+ 0.27 



— 



-1.31 



+ 0.274 sin ( Z — tt) 

 — .238 sin ( V — ■k) 



— 



5 



+ 



11 



— 5 cos 

 — 10 cos 



V 



l- 



") 



(1979.72) 



(1955.50) 



+ 4. 365 sin ( 2Z'— l — v) 



(- 



1141) 



(- 



1127) 



+ 43 cos 



21' 



— l- 



- n) 



(+69.86) 



(+ 



68.73) 



+ 9 .563 sin ( 3 Z' — 2 Z — ir) 



(+ 



693) 



(+ 



663) 



+ 58 cos 



3Z 



— 21- 



tt) 



— 1.78 



- 



-1.78 



— 1.721sin( 4Z' — 3Z — tt) 



— 



28 



— 



27 



— 27 cos . 



4Z' 



— 3Z- 



tt) 



— 0.33 





-0.59 



— .375 sin ( 5 Z' — 4 Z — tt) 



— 



7 



_ 



5 



— 7 cos 



5Z' 



— 4Z- 



tt) 



— 0.12 





-0.29 



— 0.134 sin ( 6Z' — 5Z — tt) 



_ 



3 



_ 



1 



— 2 cos 



6Z' 



— 5Z- 



■k) 



— 0.06 







— .057 sin ( 7 Z' — 6 Z — tt) 



_ 



2 







— 2 cos 



7Z' 



- 6Z- 



n) 



— 0.04 







— .022 sin ( 8 Z' — 7 Z — tt) 











— 2 cos ( 



8Z' 



— 7Z- 



"") 



— 001 







— .009 sin ( 9 Z' — 8 Z — tt) 



















May, 1865. 



