THREE CASES DISTINGUISHED.

441

the normal force lie somewhat beyond the quadratures, on the sideof the opposition, and do not undergo any very material changeof situation with the contraction of the disturbing orbit.

(717.) The third case is that in which the diameter of the dis-turbing interior orbit is less than half that of the disturbed. Inthis case there are only two points of evanescence for eitherforce. Those of the tangential force are the syzygies. The dis-turbed planet is accelerated throughout the whole semi-revolutionfrom conjunction to opposition, and retarded from opposition toconjunction, the maxima of acceleration and retardation occurringnot far from quadrature. The neutral points of the normal forceare situated nearly as in the last case; that is to say, beyond thequadratures towards the opposition. All these varieties the studentwill easily trace out by simply drawing the figures, and resolvingthe forces in a series of cases, beginning with a very large andending with a very small diameter of the disturbing orbit. It willgreatly aid him in impressing on his imagination the generalrelations of the subject, if he construct, as he proceeds, for each

Fig. 101.

■n

/ N

E

case, the elegant and symmetrical ovals in which the points N andL [fig. art. 675,) always lie, for a fixed position of M, and ofwhich the annexed figure expresses the forms they respectivelyassume in the third case now under consderation. The secondonly differs from this, in having the common vortex m of bothovals outside of the disturbed orbit A P, while in the case of anexterior disturbing planet the oval mL assumes a four-lobed form ;its lobes respectively touching the oval m N in its vertices, andcutting the orbit A P in the points of equidistance and of tangency,( i. e. where j\I P S is a right angle) as in this figure.