Prof. Cayley on Differential Equations and Umbilici. 373 



probability that these subchlorides contain half as much chlorine 

 as the chlorides from which they are prepared. But the oxides 

 and chlorides of the alkaline metals are not isoxides and isochlo- 

 rides, bat semioxides and semichlorides, and the atomic weights 

 of the five known alkaline metals (as well as that of thallium) must 

 be divided by 2. These subchlorides are then quadrantichlorides, 

 to which quadrantoxides probably correspond; the latter will 

 probably, however, be difficult of preparation. The group of 

 quadrantoxides might then be called a large one*. 



LIII. On Differential Equations and Umbilici. By A. Cayley, 

 Sadlerian Professor of Pure Mathematics at Cambridge -\. 



I. 

 CONSIDER the integral equation 

 ^ As 2 + 2B*-fC = 0, 



where z is the constant of integration : the derived equation is 

 n = (AC + A'C - 2BB') 2 - 4( AC - B 2 ) (A'C - B' 2 ) 

 = (CA'-C'A) 2 -4(AB'-A'B)(BC'-B'C)=0; 



which are reddish brown in colour. The blue coloured cubes consist only 

 of chloride of sodium (with a very small quantity of sulphate of soda) ; the 

 colourless and the reddish brown ones contain a great deal of chloride of 

 potassium ; the colourless ones 2 atoms chloride of potassium and 1 of 

 chloride of sodium, hence 73 per cent, chloride of potassium. 



A similar relation prevails in the case of the blue rock-salt of Kalucz in 

 Galicia. In these also, blue coloured cubes trench on perfectly colourless 

 ones. The latter consist of chloride of sodium only, the former of pure 

 chloride of potassium without any mixture of chloride of sodium. At the 

 same time there are cubes which are of a faint brownish colour, and yet 

 contain no chloride of potassium, but consist of pure chloride of sodium. 



Not in all cases do the colourless cubes, which lie close to the blue ones, 

 contain much chloride of potassium, or consist of it, while the blue salt 

 contains none. Bluish-coloured rock-salt of Hallstadt was found to be 

 quite free from chloride of potassium (it sometimes contained very small 

 quantities) ; but the adjacent colourless cubes consisted also of chloride of 

 potassium. Something similar was found in the case of a feebly blue 

 rock-salt from Wieliczka. 



The bluish-coloured rock-salt, like the colourless, forms a perfectly 

 colourless solution which has no alkaline reaction. All kinds of the blue 

 rock-salt, even those which are tolerably intensely coloured, like that of 

 Kalucz, give rise to no disengagement of hydrogen when dissolved. 



* The experiments described in this paper have been made by Dr. Fin- 

 kener, who has devoted himself to this research with unusual zeal. It 

 was only by his skilful arrangement of the apparatus that it was possible 

 to prepare the very oxidizable quadrantoxide of copper; and I confess that 

 without his help I could not have obtained the results I have described. 



t Communicated by the Author. 



