1880.] History of Planet and Single Satellite. 



255 



Fig. 25. Plasmodium {Cancer Pagurus). 



Fig. 26. Corpuscles of Asteracanthion vulgare, freshly drawn. 



Figs. 27 — 28. Union of a group of corpuscles of Asteracanthion vulgare. 



Fig. 29. Portion of a plasmodium produced by the union of the finely granular 

 corpuscles of JEcJiinus sphcera, showing distinct endoplasm containing 

 the coarsely granular and the coloured corpuscles, and ectoplasm sending 

 out filamentous pseudopodia, which unite with those of free corpuscles. 



Figs. 30—33. Phonergates vorax, from " Zeitsch. f. Wiss. Zool." Bd. XXX. 1878. 

 Taf. II, figs. 54-57. 



All the figures drawn with Yerick, Oc. 2, Obj. 7. 



IV. " On the Analytical Expressions which give the History of 

 a Fluid Planet of Small Viscosity, attended by a Single 

 Satellite. 7 ' By G. H. Darwin, F.R.S. Received March 6, 

 1880. 



In a series of papers read from time to time during the past two 

 years before the Royal Society, I have investigated the theory of the 

 tides raised in a rotating viscous spheroid, or planet, by an attendant; 

 satellite, and have also considered the secular changes in the rotation 

 of the planet, and in the revolution of the satellite. Those investi- 

 gations were intended to be especially applicable to the case of the 

 earth and moon, but the friction of the solar tides was found to be a 

 factor of importance, so that in a large part of those papers it became 

 necessary to conceive the planet as attended by two satellites. 



The differential equations which gave the secular changes in the 

 system were rendered very complex by the introduction of solar 

 disturbance, and I was unable to integrate them analytically ; the 

 equations were accordingly treated by a method of numerical quadra- 

 tures, in which all the data were taken from the earth, moon, and 

 sun. This numerical treatment did not permit an insight into all the 

 various effects which might result from frictional tides, and an analy- 

 tical solution, applicable to any planet and satellite, is desirable. 



In the present paper such an analytical solution is found, and is 

 interpreted graphically. But the problem is considered from a point 

 of view which is at once more special and more general than that of 

 the previous papers. 



The point of view is more general in that the planet may here be 

 conceived to have any density and mass whatever, and to be rotating 

 with any angular velocity, provided that the ellipticity of figure is not 

 large, and that the satellite may have any mass, and may be revolving 

 about its planet, either consentaneously with or adversely to the plane- 

 tary rotation. On the other hand, the problem here considered is 

 more special in that the planet is supposed to be a spheroid of fluid of 

 small viscosity ; that the obliquity of the planet's equator, the inclina- 



