1878.] G. H. Darwin on the Tides of Viscous Spheroids. 419 



AAA 

 COS \{aft) + COS \{ftj) + COS + & 



V aft ft<y s/ <y& 



cos |Qa) _q „ 



v ua 



I have examined this equation at length in the special case n~4. 

 The tetrazomal curve, which is of the eighth degree, breaks up into 

 two factors. One factor represents the circle and the square of the 

 line gaining the points (7), (ftd), the other denotes an unicursal 

 quartic whose three double points are (a<y) and the pole with 



respect to the circle of the line joining these points. 



These results hold for the plane and sphere, and a, ft, 7, 8, &c, may 

 denote circles as well as lines. They are also true 'for conies having 

 double contact with a given conic. 



Besides the foregoing, which are manifestly extensions of known 

 theorems, the paper contains some original theorems which are also 

 extended to conies having double contact with a given conic. Thus, 

 " if a circle 2 be touched by any number of circles Si S 2 S 3 , &c, and 

 if we denote by P(V) the product of the common tangents drawn from 

 any circle S; of the system to all the remaining circles, then the 

 equations of 2 will be a factor in the polyzomal curve, 



y/s7 =5 _.v / S? =i + v / S^- &0. =0. 

 P(l) P(2) P(3) 



This theorem is, I believe, one of the most fertile in geometry. The 

 paper contains a large number of deductions from it, but I am certain 

 it is far from exhausting them. 



V. " On the Bodily Tides of Viscous and Semi-Elastic Sphe- 

 roids, and on the Ocean Tides on a yielding nucleus." By 

 George H. Darwin, M.A., Fellow of Trinity College, Cam- 

 bridge. Communicated by J. W. L. Glaisher, M.A., F.R.S. 

 Received May 14, 1878. 



(Abstract.) 



Sir W. Thomson's investigation of the bodily tides of an elastic 

 sphere* has gone far to overthrow the idea of a semi-fluid interior to 

 the earth, yet geologists are so strongly impressed by the fact that 

 enormous masses of rock have been poured out of volcanic vents in 



* Phil. Trans., 1863, p. 573, and Thomson and Tait's Natural Philosophy, edit, of 

 1867, §§ 733-737 and 834-846. 



