CRITICAL AND SUPPLEMENTARY. 215 



Equations (117) and following, combined with the hydrostatic equation (3), 



du u^ 



and (122) gives 



du 

 from combination of which comes 



-^=^ y=e z=uQ (125) 



dy z^ 



This equation determines a family of curves in the y-z plane, of which the 

 one required for the present purpose passes through the origin, at which 

 the 2-axis is an inflectional tangent. In the neighborhood of the origin 

 this curve is given by the expansion 



y-k^+y^w6''+ ■■■■ (^2«' 



with coefficients all positive. The arc required is that lying in the quad- 

 rant where y and z are positive. j 3 

 In the half-quadrant where y>z, the inequality -i->2^ gives y>-i^, 



which shows that the curve reaches the line y = z at some point {a, a) 

 such that 0<a<-y/3. Crossing this line, with tangent parallel to the y- 



axis, the curve passes into the half-quadrant where y^z, where the in- 



dz (X — 1/ 

 equality -j-< — i — gives 2^<a^ — 2(,v — a)^, which shows that the curve must 

 ay z 



cross the y-axis, with tangent parallel to the 2-axis, at a point (2 = 0, t/=i9) 



a^ 1 /— 



such that ^<a-\-—F= or j3<a(l +^^/6). The curvature does not change 

 ^2 ^\ 



sign in this quadrant, since the differential equation gives 



d'z 

 dy' 



which is constantly negative where y and z are positive. This part of the 

 curve is therefore a simple arch concave to the y-axis and crossing per- 

 pendicularly at j/ = and at j/=j8. 



This means that there is a finite critical mass which is reached only 

 when the function (i){u) becomes zero and consequently the density at the 

 center infinite. The corresponding value of u is then infinite, but the 

 radius is finite, since the density must be everywhere greater than p^. 



This result may be compared on the one hand with the case n = -^, where 



the limiting radius is reached while the mass and central density are both 

 finite; on the other hand with the case n = l, where the limiting radius is 

 approached only asymptotically as the mass increases indefinitely. 



From the rejection of the hypothesis last developed it appears, there- 

 fore, that under the law of compressibility which would result from any 



, = -[3(2-y)'-2(2-y)-h2*]^2^ 



