File:Some Problems Illustrating the Forms of Nebulae (1915) (14585890140).jpg

Identifier: philtrans02048693 (find matches)
Title: Some Problems Illustrating the Forms of Nebulae
Year: 1915 (1910s)
Authors: Walker, G.
Subjects: Proceedings of the Royal Society of London Philosophical Transactions of the Royal Society
Publisher: Royal Society of London

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- , ,% r=r (J 3#2 cosh2w# has a solution z = A tanh w#. Hence the equation may be written* coth nd 777: tanh2 nd ^.z coth nO = a2^ e2df2>60 06 06 and the particular integral which we require is z == a2 tanh ?&0 coth2«?i0 td,nhn6 ^f2e2B d6 dd.* Cf. Boole, Differential Equations, p. 205. Some Problems Illustrating the Forms of Nebulce. 417 For uniform rotation/ = co, and then z = 2 a2m2 tanh nd coth2 n6 tanh n6e2e d6 dQ. In the ease n = 1 this gives <X2&)2 3 = 41og(y + l) + (y-i)(y+i) 1-,71og(y+1) dy and for w <X2ft)2 ^ = 2 (tan 1y- V 1 + yV+ $*+!)J (l-ytan-lyjrfy 2 l \where ?/ = r2/<x2. These curves are shown to scale in figs. 5 and 6, and we note that in bothcases z = 0 when r = 0,is everywhere positive, and finally becomes infinite andproportional to r^/a2 when r is great. 109 FIG.5
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The general effect of the rotation is then to increase the density from whatwould obtain in the statical case, and by increasing proportion as r increasesfrom the origin, but even if it were permissible to pass to large values of rthe density would still diminish to zero. We cannot, however, extend our 418 Mr. G. W. Walker. solution to large values of z, but we may say that the tendency of the rotationis (e.g. when n = 2) to make the density rise more steeply to its maximum

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