IN THE MOTIONS OF THE EARTH AND VENUS. 
123 
and the differences between these are almost exactly proportional to the times. 
They show that the secular motion ought to be increased by 4", 8 (the preces¬ 
sion being supposed the same as in the application of Maskelyne’s catalogue); 
and then the application of the inequality investigated in this memoir will give 
correctly the Sun’s mean longitude. 
It appears, however, that the inequality in the motion of the perihelion given 
by this investigation, will not account for the anomalies in the place of the 
perihelion given in my paper referred to above. 
Thus terminates one of the most laborious investigations that has yet been 
made in the Planetary Theory. The term in question is a striking instance 
of the importance to which terms, apparently the most insignificant, may some¬ 
times rise; and the following remark will show the magnitude of the errors 
which might, under other circumstances, have arisen from the neglect of this 
term. If the perihelia of Venus and the Earth had opposite longitudes, and if 
the line of nodes coincided with the major axes, the excentricities and incli¬ 
nation having the same values as at present, the coefficient of the inequality 
in the epoch would be 8",9 ; and all the other terms would be important. A 
very small increase of the excentricities and inclination would double or treble 
these inequalities. 
I have avoided any discussion of physical theory, as little can be added 
at present to what has been done by Laplace and others. I may remark, 
however, that my expression for ^ differs from that given by Laplace ; and 
that the difference produces no effect in the ultimate result, because Laplace 
uses fndt where I have usedrctf. On this point I have only to state that, by 
adopting the expression which I have used, every formula for the longitude, 
the radius vector, and the velocity in any direction, is exactly the same in 
form for the variable ellipse as for an invariable ellipse (taking the variable 
elements instead of constant ones). If the disturbing force should at any 
instant cease, my value of s for that instant would be the true value of the 
epoch of mean longitude in the orbit which the planet would proceed to 
describe. It is precisely the object of using the method of variation of ele- 
