ASTEROID SUPPLEMENT. 9 



The want of accuracy in the mean distances of many of the Asteroids renders 

 some simple and expeditious means desirable for correcting the coefficients for 

 changes in the value of the argument a. The following formulas and tables 

 enable us to compute these changes with every possible facility and accuracy, and 

 leave nothing to be desired in the numerical solution of the problem. 



Denote by A t , B t , C t , D t , &c. the variations of the tabulated functions for a 

 change of .001 in the argument a, the logarithms of which at any value of a are 

 found in the general Tables. If A denotes a difference of the function correspond- 

 ing to a change A a = .001 in the argument, and il/is the modulus of the common 

 system of logarithms, we readily find the following formulas. 



VARIATIONS OF LOG ¥'J. 

 A log b l $ =At + 



,(,) , iMAa 



for all values of i. 



VARIATIONS OF FIRST DERIVATIVES. 

 A log a D„ J ( J = B, + (» + 2 p) *^ 



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



A log a D, J«J» = B + (2+ 2/32) ^Af . 



VARIATIONS OF SECOND DERIVATIVES. 

 A log I Dl i ( j> = C, + (t + 4 p) *±2 

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



Aloga 2 Dtb<? = C + (2+ 4 0*)^p, Aloga 2 Dl b<$ = C, + (3 + 4 £») ^ . 

 VARIATIONS OF THIRD DERIVATIVES. 

 A log a D.j. b\ = -A -f- (* ■ + " /3-) 



for all values of i except i = 0, 1, and 2, for which the formulas are 



Aloga 3 « = A+.(4+ 6/3°-) ^, Aloga"rfi ( ?= A + (3 + 6 /3*) ^ . 



. ''a ' v ' ' a 



Aloga 3 J Df6'|»=A+(4 + 6^)^- a . 



VARIATIONS OF FOURTH DERIVATIVES. 

 A log a 4 ^ »<?= B, + (i + 8/3^) ^ 



for all values of i except i = 0, 1, 2, and 3, for which the formulas are 



A log a 4 .^=£0+ (4 + 8 02)^, A log a* 2)*^= £,+ (4 + 8/3*)^. 



Aloga 4 D^ ( !'= £, + (5 + 8/3*)^, Alog.a 4 i> 4 &f=£ 3 +(5 + 8^)^, 



VARIATIONS OF FIFTH DERIVATIVES. 

 A log a 5 D* J'? = P, + (« + 10 /32) ^Af 



for all values of i except i = 0, 1, 2, 3, and 4, for which the formulas are 



