THEORBITOPURANUS. 41 



The multipliers on the left are each one-half the coefficient cos jij in the ex- 

 pression for - , , and each product is placed in the two columns corresponding 



respectively to N-\- jg and N — jg. 



All the derivations of i^ necessary in the computation of the perturbations of 

 the first order are given in the following tables. First we have tlie values of 

 D'tR obtained by direct differentiation, as indicated in the preceding formuhe. 



Next we have ' and , , obtained bv the formuhe (7) and (4',^). The products 



ov dp ' V / V / 1 



— — by - , " find of ~^— by -^, being formed in the simple wav just pointed out, 

 (9v •' (/t dp (H i. V J 1 , 



and with the values of the component factors just given, their sum is next shown. 

 This sum should agree accurately with I)\R. The discrepancies are shown in the 

 next two columns. The only apparently large discrepancy is found in the argu- 

 ment og' — 51. It probably arises from the incompleteness of the computation of R 



and ^--, so far as they depend on this argument. As the entire term does not 

 dv 



amount to 0".01, I have not sought to correct it. 



The great value of this check arises fi-om the fact that it gives a complete con- 

 trol of the correctness of the development of the perturbative function, ab initio, 

 since the two valves of D'lR are derived from different terms of that development. 



c>R 



It also controls all the computations except that of . This quantity bcin"' 



multiplied by quantities of the order of the eccentricities in the second value of 

 D'fR, an error in its value will produce a discrepancy of only ^\ its own amount in 

 D'lR, and may therefore be overlooked. The derivative in question must there- 

 fore be checked by a complete duplicate computation. 



In the column next following are given the integrating fiictors r, for which the 

 expression is 



n 1 



tU -\- 171 



For each value of i' the values of j' are therefore the reciprocals of a series of num- 

 bers in arithmetical progression, the common difference being unity. 



April, 1873. 



