no 



metrical functions of a and d ; and the different quantities C are 

 of course connected by equations of condition similar to the others. 



The general development of it, given in the third volume of the 

 Mecanique Celeste, extends only to the terms depending on the 

 squares of the excentricities. Burckhardt carried the development 

 much further in the Memoires de Vlnstitut for 1808 ; but Professor 

 Airy's formulae are not immediately comparable with his, on account 

 of the employment of C instead of b. The formulae of Burckhardt 

 include however only 6 out of 12 of Professor Airy's coefficients, in 

 consequence of the omission of terms depending on the inclination, 

 by the former mathematician. 



In the expansion of R, the terms which proceed according to the 

 powers of the excentricities and of the sine of half the inclination 

 (e, e\ f\ involve cosines of the multiples of V, T, and V — T, 

 (V and T indicating the mean motions of Venus and the Earth with 

 the addition of certain constant quantities.) By expanding also the 

 variations of the radii vectores, which occur in these terms, according 

 to powers of e, e\ the cosines of multiples of the arcs V and T again 

 enter these terms. Hence the series will finally contain the pro- 

 ducts of the cosines of the three kinds of arcs, namely, multiples of 

 V—T, V and T; which, as is well known, may be expressed in 

 terms of the cosines of their sums and differences. These last arcs 

 will produce various combinations of the form pV — q T. Of these 

 we are to select those in which the arc is 13 V — 8 T; for such 

 terms will, in the calculation of the perturbations, be divided by very 



small quantities (namely, either -i- or —J— ), and may thus pro- 



duce results of appreciable magnitude. 



It may serve to assist us in forming a judgement concerning the 

 place of Professor Airy's memoir among the laborious calculations 

 of physical astronomy, if we compare it with the investigation of the 

 great inequality of Jupiter and Saturn, as originally given by La- 

 place {Mem. Acad. 1785), the undertaking to which it has the closest 

 analogy among such researches*. 



The number of terms arising from the combination of the three 

 arcs above mentioned, V — T, V, and T, which give 13 T — 8 V, is 

 considerable ; but many of them are rejected on account of their co- 

 efficients going beyond the 5th order. The following six are re- 

 tained and have their value calculated : those in which V — T, V, T 

 are respectively 8, 5, ; 9, 4, 1 ; 10, 3, 2 ; 11, 2, 3 ; 12, 1, 4 ; 

 13, 0, 5. In the Jovi-Saturnian inequality, if V and T still refer to 

 the mean motions of the exterior and interior planets, four combi- 

 nations are taken, namely, those in which V — T, V, T are re- 

 spectively 2, 3, ; 3, 2, 1 ; 4, 1, 2 ; 5, 0, 3. 



The number of terms of the calculation depends also upon the 



* The comparison is here made with the investigation of the principal 

 term only of the Jovi-Saturnian inequality, as the most celebrated and most 

 analogous, not as the most laborious or most recent, of similar investigations. 



