114 JUPITER AND SATURN. 



To test the accuracy of these figures, use has been made of the 

 equation 



D™(d+,-'1=^)&,,.=«(D + i)"(d + .+'^)&,_i,„. 



This affords a complete test, because it connects even and odd 

 -values of D'" h„ whilst the method of calculation keeps these quite 

 separate. 



In particular we have (D +3) 6,,, = o (D +7) ij,,, or 13-2316 

 = X 242693 =13-2316, thus D64., is correct, hence D^J.., is 

 correct, and its correctness implies that D'6„.i and all lower orders 

 are correct. 



Then (D"' +8 D^) 6,,.3 =a (D^ + 14 D' +46 D' +64 D^ +41 D +10) ^8,3 

 = 12474-15 being exact, ensures that all the quantities to D"6s,i are 

 exact. 



It is possible that the value of a adopted (log = 97365540) 

 may require alteration. If so, as already stated, it will not be 

 necessary to recompute the coefficients de novo, because we have 



For (a + Aa) For o 



D- h,„ =D"' &,,„ + log. (i +^) D'"+' ^-.» +i log- (^ +^')1'-'^'"^' ^'v.+&c. 



(see M.N.R.A.S. LXIX, p. 647, where, however, p\ has been mis- 

 printed for !//>!). 



If the correction to log a is less than 0-0000434 the first and 

 second right-hand terms need only to be retained, to secure accuracy 

 to seven places of decimals. It will be understood that log 



ii-\ — ^ j is a natural logarithm. 



If A log o = ± 0-0000434, a value which can hardly be exceeded, 

 we have 



log \i ± — ") — ± o-oooiooo 



and 



\ log(i± — j=-i- 0-0000000,05 



or more generally 



log (i± — ^ = ± 2-3026 A log o. (0-36222). 



It is to be remarked that these incremental factors depend on 

 S a and are independent of the actual value of a. 



With these formulae, a comparison with Le Verrier's values of 

 the coefficients becomes easy and one or two examples may be 

 .given. As Le Verrier used log = 9-7367408, we have A log a 

 = +0-0001868 and 



log / I +-^J =0-000,4301,2 (6-63359). 



|[ ditto ]- = 0-000,0000,9 (2-966) 



Then c D6„, , } 



L.V.'s b'"^ = 2-1801414+ (0.4407] X 0-0004301 ) 

 + __^895 = < 



2-180331 in perfect agreement. 



