ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 589 
radius-vector of the disturbed body is Jess or yreater than that 
of the disturbing. If both cases should occur with two heavenly 
bodies, both methods must be connected with each other, and 
their results be employed alternately ; for example, in the case of 
Encke’s comet and the earth, and so on. In the present trea- 
‘tise the author only carries out at length the case in which the 
‘radius-vector of the disturbed body is less than that of the dis- 
rbing, and does but briefly point out the treatment of the other 
se. He reserves the full explanation of the latter till he shall 
have computed an example in illustration of it. 
_ The case here treated of fully is that to which the most con- 
siderable disturbances of the bodies of our solar system belong, 
hich move in very eccentric and considerably inclined orbits. 
e comprehends in it namely the disturbances which the four 
“small planets, the comet of Encke, and the comet of Biela 
undergo from Jupiter, Saturn, and Uranus. Also, on grounds 
which will afterwards be discussed, he only enters substantially 
“on the consideration of those disturbances which Halley’s comet 
undergoes from those planets. 
_ Asa first example of his method, the author gives the calcu- 
lation of the disturbances which Encke’s comet experiences from 
Saturn, and moreover informs us that he has carried out the 
calculation of the disturbances produced by Jupiter so far as to 
arrive at a view of the nature of the result. According to this, 
the disturbances contain scarcely a greater number of terms than 
ose produced by the sun in the motion of the moon; but, in 
the former disturbances, no coefficients result so great as in the 
latter. While the greatest coefficient in the lunar disturbances 
ounts to nearly 4470", the greatest in the disturbances of 
Backe’ s comet by Jupiter amounts only to 2480". The secular 
variation of the eccentricity is small, but the yearly motion of 
_ the line of apsides of this comet produced by Jupiter amounts 
to more than half a minute. The disturbances which this comet 
undergoes by Saturn are, when developed, as follow :— 
Let g! be the mean anomaly of Saturn corrected by the great 
inequality ; 
uw ...the eccentric anomaly of Encke’s comet ; 
t ...the time, whose unit is a Julian year; 
noz..the disturbance of the mean longitude ; 
w...the corresponding disturbance of the hyperbolic 
logarithm of the radius vector ; 
