lviii THE THEORY OF THE LONG INEQUALITY 



Suppose now a conjunction to take place when P is at perihelion. Since A has just 

 completed a revolution the mean angular velocity is reduced to its minimum value ; there- 

 fore P will be continually in every revolution dropping behind its undisturbed elliptic 

 place. But, as A is also very slowly revolving from a to b, the effect of disturbing 

 force is to restore the mean angular velocity to its undisturbed value ; and this value is 

 attained when A has arrived at b. As A passes from b to a the mean angular velocity 

 is increasing beyond its undisturbed value, and therefore the planet is regaining the lon- 

 gitude it had lost : and when A arrives at a the disturbance in longitude is reduced to 

 zero. Therefore throughout the former half of the revolution of A from the perihelion 

 the planet P is behind its undisturbed place, and similarly it is before that place through- 

 out the latter half. 



65. The fluctuation of distance of P from S produces an analogous alteration in the 

 tangential force. That part of the force which depends upon the argument ZPSP is 

 smaller when SP is smaller, and vice versa. While P is moving from b' to A (fig. 4) 

 the circular tangential force is accelerating, and SP is less than its mean value. P is 

 therefore acted on by a less accelerating force than it would be if its orbit were circular 

 (radius = mean distance) : therefore the additional tangential force is retarding. While 

 P moves from A to b the circular tangential force is retarding, and SP less than its 

 mean value : therefore the additional force is accelerating. Similarly it is i"etarding from 

 b to A' and accelerating from A' to b'. Therefore there is from this cause also a pre- 

 ponderating force, which is retarding while A revolves through aba and accelerating while 

 it revolves through a'b'a, coming to a maximum when A is at b and b' and vanishing 

 when it is at a and a . It is therefore always of the same sign as the force investigated 

 in the last article, and concurs with it in producing the same perturbation. 



66. We have hitherto considered only that alteration of the tangential force which 

 is due to the deviation in the position of P. But the tangential force in the ellipse will 

 also differ from that in the circle on account of the difference of the direction in which 

 the disturbing forces must be resolved. The forces which are normal and tangential in 

 the circle (and which we have hitherto treated as if they were normal and tangential in 

 the elliptic orbit too) are radial and transversal in the ellipse. These forces must there- 

 fore be both resolved along the tangent in order to obtain the correct value of the tan- 

 gential force. We might have supposed the disturbing forces to be always resolved 

 along the normal and tangent, but in that case we could not have determined the effect 

 due to the deviation of P from its mean place separately from the effect produced by the 

 change in the direction of the tangent. 



First, the radial force will contribute a portion of the tangential force. That part of 

 it which depends upon the argument 2PSP acts outwards while P moves through B'AB 

 and inwards during the other half of the revolution. When P is anywhere in the arc 

 aAB (fig. 5) the radius vector makes an acute angle with the tangent which is turned 

 towards the perihelion a. The radial force, therefore, when resolved along the tangent 

 produces an accelerating tangential force which does not exist in the circular orbit. When 



