THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 95 



For the root f/ 5 =2".7706502, we get, 



. 0.6382110 Af/r 0.1345391 Ar/r 318.2862 „ m 



**=+ 10 o W ; y>= w — xr-, 2 5 =— 1()10 -Nf*. 



Whence / i 5 =101° 54' 14".5 ; log. iV 5 ' r =7.31 15881. 



N b =+0.00061393; ^'"=+0.00204922; 



iV 5 ' =+0.00059396 ; NJ =+0.0()lS554s ; 



JV 5 " =+0.00062031 ; iV 5 "' =+0.0300090 ; 



JV/'=+0.00082154 ; iV 5 r/ =— 0.00288234. 



For the root r/ =3".722656, we get, 



0.4577640 . r/r , 0.8595159 xr . F . 22.63367 XTtn 



X « = " 10" ' u ; z/u=+ W° — N * ' 2 * = + io' u ° * 



Whence (3 B =28° 2' 20".5; log. ivy "=8.6337215. 



JV =+0.0245381 

 iV G ' =+0.0165876 

 JV 6 " =+0.0163169 



JV '"=+0.0187473 



For the root r/ 7 =22".466817, we get, 



N/ r =+0.0430251; 

 N a r =+0.0339716; 

 iV 8 "=— 0.0466232; 

 NJ"=— 0.0015353. 



1.009767 0.7872124 ^ _ , 81.89 131 /r2 



io'° 7 ' ^ 7 — ' io 10 ' Zt — ' 10 10 ' 



Whence /3 7 =307° 56' 23". 7 

 JV 7 =—0.0000974 

 N 7 ' =+0.0003164 

 N 7 " = —0.0023759 

 iV 7 '"=— 0.0150168 



log. JV/ r =8. 1940957. 



JV/ F =+0.0156349 

 N 7 r =—0.0483516 

 N 7 rT =+0.0018091 

 JV 7 r/, =+0.0001364. 



42. We have thns obtained the values of all the constants, corresponding to the 

 separate variations of the planetary masses. If we now subtract the values of the 

 constants which correspond to the assumed masses from the values which result 

 from the supposition that each planetary mass receives in succession a finite incre- 

 ment, and divide the difference of the constants by the supposed increment of 

 mass, and connect together the different results, we shall have the following system 

 of equations for the determination of the constants which correspond to any other 

 assumed finite variation of the masses. The unit of the coefficients of [i, p, ft", &c, 

 in the values of N, N', N", &c, N lt iV"/, N{, &c, are the seventh decimal place of 

 these coefficients. 



