. Some Problems Illustrating the Forms of Nebulae . two oval curves about F and F, which are the foci of the original ellipsesin the transformation. F and F7 are singularities at which p would beinfinite. The general character of the loci is not altered by giving f0 somefinite value. For comparison fig. 4 shows on the same scale the result ofputting n = 2. The equation is __ joi 1 sin2?; + l — cosh2£, p cosh22 £ and the curve drawn is for P/pi = 1. It is seen to be rather flatter than thecorresponding curve for n = 1. This case then, as a whole, has some analogy in the case of a nebula withtwo
. Some Problems Illustrating the Forms of Nebulae . two oval curves about F and F, which are the foci of the original ellipsesin the transformation. F and F7 are singularities at which p would beinfinite. The general character of the loci is not altered by giving f0 somefinite value. For comparison fig. 4 shows on the same scale the result ofputting n = 2. The equation is __ joi 1 sin2?; + l — cosh2£, p cosh22 £ and the curve drawn is for P/pi = 1. It is seen to be rather flatter than thecorresponding curve for n = 1. This case then, as a whole, has some analogy in the case of a nebula withtwo fundamental equal nuclei. Some Problems Illustrating the Forms of Nebidce. 415 Numerous other cases may be worked out, and it is simply a question ofdetail in assigning forms to the conjugate functions and -v/r. But perhaps p//>*\ -5. these three cases are sufficient to indicate the generality of the method, andthe remarkable interest of the forms that can be obtained. I pass to the consideration of an important related problem. The material of an actual nebula may be moving and not at rest, and inparticular it may be rotating. As the rotation, if it exists at all, is veryslow, the divergence from the statical equilibrium in such a case would beextremely small, so that the statical solutions are themselves of value. Itis, however, of interest to know what the effect of motion may be. The simplest case we can consider is the final state in which the materialrotates about an axis like a rigid body, with steady angular velocity a>.There the problem reduces to a statical one, when we take axes rotatingwith the material and add to the gravitational forces the centrifugal the case of a gas such a rotation, however small, can hardly be expectedto extend indefinitely, and we should rather expect the motion to
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