OF URANUS AND NEPTUNE. 



lix 



P is between B and a the radial force acts inwards and the angle between the radius 

 vector and the tangent is still acute, therefore the additional force is retarding. Similarly 

 through the arc a'A'B" the angle between the radius vector and the tangent being always 

 acute towards perihelion, and the radial force acting inwards, the additional force is ac- 

 celerating ; while through B'A, the radial force acts outwards and therefore the additional 

 force is retarding. Here, then, we have an excess of positive force which tends to produce 

 a disturbance of a directly opposite character to that previously examined and which goes 

 through its changes in the same time. 



67. Again, the circular tangential force (or transversal force) must be resolved along the 

 tangent to the elliptic orbit, and the difference which this resolution produces may, as before, 

 be represented by supposing a small force to act in addition to the circular tangential force. 

 Let \js equal the angle between the normal and the radius vector, and let T equal the 

 circular tangential force depending upon 2PSP. Then the additional force is (cos \|y — 1)7*. 

 Now T is negative through the arc Ja'A', therefore the additional force is positive ; and 

 T is positive through the arc A'aJ, therefore the additional force is then negative : and 

 it vanishes at aAd A'. The additional force in the arc aA is equal and opposite to that 

 in the arc A' a, and the force in the arc ABa is equal and opposite to that in the arc 

 A'B'a ; this force therefore completely compensates itself in one revolution of P, and 

 therefore produces no part of the long inequality. 



68. The momentary change of major axis is greater, cceteris paribus, when the linear 

 velocity of the disturbed planet is greater. Between A and b (fig. 6) the velocity is above 

 the mean value, and therefore the effect of the retarding force is greater than it would 

 have been if the velocity had been uniform. Through the arc ba'A' the velocity is be- 

 low the mean, and the retarding force produces a less effect. Through A'b' the velocity 

 is still below the mean value; therefore the accelerating force produces a less effect: while 

 through b'aA it produces a greater effect because the velocity is above the mean value. 

 Upon the whole, therefore, the retarding force produces a less effect and the accelerating 

 force produces a greater effect than in the circular orbit. There is therefore from this 

 cause a long inequality of opposite character to that investigated in articles (64) and (65).* 



" The terms explained in Arts. 64—68, may be traced in 

 the following manner. Let 4>, <f>' be the longitudes of PP' in 

 their circular orbits, a the radius of P's orbit, and V its velo- 

 city = an. Then, 



d(R) _ 

 dt 



dR dd> dR „ 



-3— -f- = —7- V = — tan force x velocity. 



d<j> dt <id(j> 



Now let the orbit of P be supposed to become elliptical, while 

 that of P' remains circular, and let Or be the long, and rad. 

 vector of P. Then, 



d(R) 

 dt 



dRdB dR dr 

 ' d9 dt + dr dt 



_ /dR_ rdfl dR dr\ ds 

 " \rdO ds + drds) It' 



ds 

 But 6 = <t> + Sip, and r = a + ia, and — = V + IV, where 



$<f, = 2e sin ft ta = - ae cos ft i V = nae cos /3, omitting terms 

 above the first order ; 



dR £R 



rdd ~ adiji adcp* 



dR 



rdd 



d9 

 dr 



da \ad<pj 



■■ cos L between normal and rad. vector = 1 - } 2 e 1 sin 2 ft 



i sin of same L = c sin/3; 

 dR dr dR 



dr ds da 



e sin ft 



H2 



