201 



The problem, in a triangle to describe three circles each of them 

 touching the two others and also two sides of the triangle, has been 

 termed after the Italian geometer, b)^ whom it was proposed and 

 solved, Malfatti's Problem. The problem to which the author refers 

 as Steiner's extension of Malfatti's Problem, is as follows : " To de- 

 termine three sections of a surface of the second order, each of them 

 touching the two others, and also two of three given sections of the 

 surface of the second order," a problem proposed in Steiner's me- 

 moir ' Einige Geometrische Betrachtungen,' Crelle, t. i. The geome- 

 trical construction of the problem in question is readily deduced 

 from that given in the memoir just mentioned for a somewhat less 

 general problem, viz. that in Vvdiicli the surface of the second order 

 is replaced by a sphere ; it is for the sake of the analytical develop- 

 ments to which the problem gives rise that the author proposes to 

 resume here the discussion of the problem. 1 h.e following is an 

 analysis of the present memoir ; — 



§ 1. Contains a lemma which appears to the author to constitute 

 the foundation of the analytical theory of the sections of a surface of 

 the second order. 



§ 2. Contains a statement of the geometrical construction of 

 Steiner's extension of Malfatti's problem. 



§ 3. Is a verification, founded on a particular choice of co-ordi- 

 nates, of the construction in question. 



§ 4. In this section, referring the surface of the second order to 

 absolutely general co-ordinates, and after an incidental solution of the 

 problem, to determine a section touching three given sections, the 

 author obtains the equations for the solution of Steiner's extension 

 of Malfatti's problem. 



§ 5. Contains a separate discussion of a system of equations, in- 

 cluding as a particular case the equations obtained in the preceding 

 section. 



§§ 6 & 7. Contain the application of the formulee for the general 

 system to the equations in § 4, and the development and completion 

 of the solution. 



§ 8. Is an extension of some preceding formulae to quadratic 

 functions of any number of variables. 



3. " On the Tides, Bed and Coasts of the North Sea or German 

 Ocean." By John Murray, Esq. Communicated by George Rennie, 

 Esq., F.R.S. Received March 20, 1852. 



The author commences his paper by remarking that great similarity 

 of outline pervades the w-estern shores of Ireland, Scotland and 

 Norway, and then observes that the great Atlantic flood-tide wave, 

 having traversed the shores of the form.er countries, strikes with 

 great fury the Norwegian coast between the Lafoden Isles and Stad- 

 iand, one portion proceeding to the north, while the other is de- 

 flected to the south, which last has scooped out along the coast, as 

 far as the Sleeve at the mouth of the Baltic, a long channel from 

 100 to 200 fathoms in depth, almost close in shore, and varying 

 from 50 to 100 miles in width. After describing his method of con- 



