10 ASTEROID SUPPLEMENT. 



A log a 5 it b l V = F + (6 + 10 /*>) — , A log a 5 2j£ »<? = P 2 + (6 + 10 |8») ^ , 



A log a 5 DS ^ = F, + (5 + 10 ft £*? , A log a 5 2)5 j(» = ^ + (5 + 1Q ^ ^Aa 



a a 



A log a 5 D* &<£> = F 4 + (6 + 10 /9») ^A_ a . 



VARIATIONS OF LOG a¥'?. 

 A log a jf = G, + (i + 1 + 4 /3») ^^ 



for all values of »". 



VARIATIONS OF FIRST DERIVATIVES. 



MAa 



A log a 2 D„ if == Hi + (i + 1 + 6Z? 2 ) 



for all values of i except i = 0, for which the formula is 



Aloga 2 I) B if=Jff + (3 + 6,i»2)^ 



VARIATIONS OF SECOND DERIVATIVES. 



MAa 



A log a Bl J ( P = Ki + (i + 1 + 8 /3 a ) 



for all values of i except i = 0, and 1, for which the formulas are 



A log « 3 Bl b ( f = JT„ + (3 + 8 /*>) — , A log a 3 Df b^ = Z, + (4 + 8 /*») ^f , 



for all values of i. 



VARIATIONS OF LOG a 2 6^. 

 Aloga 2 bf = I, + (» + 2 + 8 /*>) — 



It will be observed, that, in all these variations, simple multiples of and 



2 yS 2 occur. If then for a change of 8 a in the argument we compute the terms 



So and 2/3 2 8a, it will only be necessary to multiply them by 1, 2, 3, 4, 



&c. to have the variations of all the coefficients arising from these terms. The 

 other terms, A t 8 a, B t 8 a, C t 8 a, &c, of the variations are as readily found, since 

 the logarithms of A^ B t , C { , &c, which are given in the general Tables, are all to 

 be increased by the same constant log 8 a. If, besides, we have tables from which 



and 2 /3 2 can be taken for any value of a by simple inspection, we shall 



be able to compute the variations of all the coefficients needed for any Asteroid in 

 a few minutes. 



Such are the Tables given on the following pages. The unit place corresponds 

 to the fifth decimal place of the logarithm to be corrected, while the unit place of 

 ■^i) B\, C ; , &c. corresponds to the last decimal in the tabulated function, which 

 is usually the seventh. 



The logarithms of 6 ( p and its derivatives, found in the following pages, have been 

 computed in duplicate to seven decimals ; but only five are printed, which is ample 

 to give all desired accuracy in the final result. 



