GRAVITATIONAL STABILITY OF THE EARTH. 207 



which is always negative. Hence w is always negative, and therefore dz/dx is always 

 negative, therefore also z is always negative and dy/dx is always negative, and 

 therefore y is always negative. Thus the equation y = has no real root other than 

 the irrelevant root x = 0, which was introduced in the process of forming the 



equation. 



Again, when v = , we multiply the left-hand member of (94) by x" and obtain the 



equation 



(23x 9 + 128x 4 -70x a + 3675) - (23x"+ 105x s + 268x 4 + 1 155x> + 3675) arVr** 1 \^*dx = 0, 



of which the left-hand member is of the order -x 6 when x is small and -x 3 when x 

 is great. We put 

 y = (23x T + 128x'-70.r 3 +3675*)e^-(23x'+105x 6 +268x 4 +1155x s +3675) 



and then we put 



\_dy_ _ Idz^ u _ Idw 



and find 



= -(512z 4 + 4552x a +3640)e* t -8832 



1 1 .1 



which is always negative. Just as before, we deduce that y is always negative. 

 It is therefore proved that the equation 



lias no real root. 



32. When n = 3, and h* = and P = 0, we have 



A, = (/, 

 where 



V. *W , y ^a" 



j- * -..4- ) 27T^c"V' 

 -A/. M. *v / y ^q" 



"-7"v 1 "T f r4~- ( } rr^-/' 



/ 4A/ a' a <*a' / y_ ^ > " 



s= 77V7" + 2T9~2~rTl H ; 2.4...2ic.(2K+5) 



A 

 ? 



3A 



/ y + i _ f" afc w - CT ...V 

 2.4...2*(2K+5)(2K+7) / 



