46 Prof. Bimsen on the Preparation of 



making in it k=¥(x, ?/, z), F denoting a function of given form. 

 We can in this way prove the theorem which is the subject of 

 this note. Let be the origin, P a point on the given surface, 

 P' the corresponding point on the first negative. By the funda- 

 mental property of derived surfaces, noticed in Mr. Hirst's 

 memoir, the normal to the surface at P will meet the line OP' 

 in its middle point M. Hence the locus of M is similar to the 

 first negative, and is had by bisecting the radii vectores of this 

 latter surface. But M is evidently the centre of a sphere touch- 

 ing the given surface, and the radius of which is equal to the 

 distance of M from the origin of coordinates. It follows, there- 

 fore, that the locus of M is found by making, in the equation of 

 the parallel surface, k <2 =x 2 + y' 2 -\-z 2 . 



Trinity College, Dublin, 

 June 13, 1862. 



With reference to the concluding part of Mr. Roberts's very 

 interesting paper, it may be noticed that his second theorem 

 relating to the parallel of the inverse surface admits of a similar 

 proof. In fact, if the inverse surface is to be touched by a sphere, 

 whose centre is at x, y, z, and whose radius is k, the primitive 

 surface will necessarily be touched by the inverse of this sphere 

 whose centre will clearly be at the point 0x, 9y, 6z, and whose 

 radius will be 6k, where has the value given above. These 

 last values, then, when put for x, y, z, k, will satisfy the condition 

 that the primitive surface is touched by a sphere, in other words, 

 will satisfy the given equation of the parallel to the primitive, and 

 transform this equation to that of the parallel to the inverse. — 

 T. A. H. 



VIII. On the Preparation of the Rubidium Compounds. 



By R. Bunsen*. 



CERTAIN varieties of lepidolite occur which, as they contain 

 more than 1 per cent, of rubidium, may with advantage 

 be employed for the preparation of the salts of this new alkaline 

 metal, as well as for that of the lithia compounds. 



I am indebted to my friend Dr. 0. Struve, of Leipzig, for 15 

 kilogrammes of a residual salt derived from the preparation of 

 lithia, which is so rich in chloride of rubidium that it may thus 

 be prepared by the pound. This residual salt consists of the 

 chlorides of sodium, potassium, and rubidium, together with 

 small quantities of chloride of caesium, and traces of chloride of 



* Translated by Professor Roscoe from Ann. der Chem. und Pharm. 

 vol. cxxii. p. 347. 



