APPLICATION OF SECTION V TO THE EARTH-MOON SYSTEM. 99 



for / = 2;r. Then it decreases to — oo in the fifth quadrant, changes sign 

 to +00 and decreases to zero at / = 3;r and to — oo again in the seventh 

 quadrant. In a general way this cycle of changes is repeated indefinitely, 

 the points where i/j becomes infinite approaching nearer and nearer to 



/ = (2n + l) ^, n being an integer. The curves y^, 2/2 are given in fig. 12. 



Both 1/1 and 2/2 being even functions of fx' the curves are entirely to the 

 right of the vertical axis. 



From the diagram it is seen that the only places where 2/1+2/2 n^ay 

 vanish are at the left of tt as at a, to the right of k, as at b, to the right of 2?:, 

 as at c, and in general to the right of nvt, n any integer. There are, in short, 

 an infinite number of determinations of //'. But when we consider that 

 the law of density is 



a' = 



we see that if 7z<ix' <2r. the density will be negative near the surface and 

 elsewhere positive; if 27r</<37r there will be a single spherical layer 

 between the center and surface where the density will be negative; and if 

 nK<ii' <{ri + l)Tz there will be n layers of negative density if n is odd, one 

 of them being at the surface, and n— 1 layers of negative density if n is even, 

 the surface density being positive. Consequently in considering such a prac- 

 tical question as the separation of the moon from the earth, in which nega- 

 tive densities would have no meaning, we need consider only the possibility 

 of a solution to the left of r. The value of //, for which t/^ vanishes, is 



-^2- 3c' 



= 142.5= 



From fig. 12 it is seen that 2/1 + 2/2 can not vanish between this point and k. 

 It is easily verified that no value of /< 142.5° will satisfy (52). For 

 example, we find the following corresponding sets of values 



l/(/)=0.58 



Consequently the smallest /x' satisfying the conditions is greater than k, and 

 the hypothesis of the separation of the moon from the earth requires, so far 

 as the factors and the law of density here considered are concerned, that we 

 assume that the surface density of the united mass was negative just previous 

 to the separation. If we had used the oblateness of the figure of the mass, 

 a still larger [x' would have been found. However, it is not impossible that 

 neglected factors may somewhat relieve the theory of these embarrassments. 

 Thus we see that when we add any of the hypotheses of an original 

 oblateness, shrinking, or different law of density singly, the difficulties of 

 the hypothesis are not relieved. 

 7 



