lloyal Astronomical Society. 383 



converging series of fractions, whose numerators contain successive 

 powers of r, the moon's radius vector, and whose denominators 

 contain different powers of the same multinomial (which, when ec- 

 centricities and inclinations are omitted, is a trinomial) that occurs 

 in computing the perturbations of the earth by Venus. Upon ex- 

 panding any of these fractions with trinomial denominator, there 

 occur terras depending on \Q(j" — \Gg\\lg" —ilg' ,axidi \Sg" —\%g' : 

 then, upon introducing the inclinations and eccentricities, the first 

 (among other combinations) will be multiplied by sin^ \ inclin. x 

 cos 2g"—'2 V (where v is the difference of longitude of node and pe- 

 rihelion of Venus), and also (in other terms) by e"'^ cos 2 g' ; the 

 second by e". e' cosg" + g' ; and the third by e'^ cos 2 g'. Each of 

 these combinations produces terms whose argument is IS g''—16 g'. 

 Then upon multiplying these terms by a power of r, since the ex- 

 pression for any power of r contains e. cos g, the product will contain 

 terms depending on IS g''— 16 g'~g. The coefficient necessarily 

 contains one of the following products of three small quantities : 

 p. sin^ ^ inclin., e . e"^, e.e'.e', c.e''^ (of which the first is the most im- 

 portant), and it is therefore extremely small; but the resulting 

 perturbation is made important by the excessive smallness of the 

 divisor introduced in integration. It is well known that the divisor 



in this case will be proportional to ( 18 -^ — 16 -^ ^ | ; and, 



*^ ^ \ dt dt dt) 



tin" 



taking for -—-, & 

 dt 



to a Julian year. 



dn" 



taking for -^, &c., the value in sexagesimal seconds corresponding 

 dt 



^ =2106G41"-3 

 dt 



^=1295977-4 

 dt 



*^ = 17179167'4 

 , dt 



whence 18^ -16^' -^ =4747"-7, 

 dt dt dt 



a quantity very small in comparison with -^. 



dt 



In this manner the greatest part of the term in question is pro- 

 duced. Other parts arise from the circumstance that, the dimen- 

 sions of the moon's orbit being slightly altered, the perturbing force 

 of the sun upon the moon is not the same as it would otherwise be. 



M. Hansen remarks that this term is remarkable as depending 

 upon higher multiples of the anomalies than have ever before been 

 considered, and as having the longest period in proportion to the 

 periodic time of the disturbed body that is yet known. 



The term depending on S g" —\^ g' arises mainly from the cir- 

 cumstance, that, the earth's motion in its orbit being different from 

 what it would have been without the perturbation by Venus, the 

 disturbing force of the sun upon the moon is not the same as if that 

 perturbation had not existed. 



