32 Prof. J. R. Young on certain Criteria of 



100 parts of cryptolite gave, — 



Peroxide of cerium . . 73*70 

 Protoxide of iron . . . 1*51 

 Phosphoric acid . . . 27*37 



102-58 

 The excess arises from the cerium having been determined 

 as peroxide instead of protoxide, in which latter form it 

 evidently exists in the mineral. Whether the oxide of cerium 

 is mixed with the oxides of didymium and lanthanium must 

 at present remain undetermined, no sure method of separa- 

 ting these oxides from each other being known. The colour 

 of the oxide obtained was dark cinnamon-brown, and evi- 

 dently contained no zirconia or thorina. The absence of this 

 latter distinguishes the mineral from monazite and edwardsite. 

 Comparative experiments made with monazite and cryptolite 

 left no doubt that they were perfectly distinct. 



In conclusion, I may mention that cryptolite is either not 

 quite insoluble in nitric acid, or the apatite of Arendal con- 

 tains another cerium mineral which is soluble; it may pos- 

 sibly be monazite. On separating cryptolite and precipita- 

 ting the saturated solution of apatite in nitric acid by an ex- 

 cess of oxalate of potash, washing and heating the precipitate 

 to redness, and dissolving the carbonate of lime thus obtained 

 in nitric acid, there remained a grayish earthy powder undis- 

 solved. Muriatic acid dissolved this powder, leaving a quan- 

 tity of fluoride of calcium. The solution, evaporated down 

 and mixed with a boiling-hot saturated solution of sulphate of 

 potash, deposited in considerable quantity a powdery, pale, 

 amethyst-coloured precipitate of the double salt of potash and 

 cerium-oxide, which, decomposed by potash, gave a cinnamon- 

 brown coloured oxide of cerium. 



VIII. On certain Criteria of Imaginary Roots of Equations. 

 By J. R. Young, Professor of Mathematics Belfast Col- 

 lege*. 



\ T page 450 of vol. xxiii. of this Magazine, I gave certain 

 -*-*- new criteria for the imaginary roots of numerical equa- 

 tions. These, in conjunction with the long-neglected criteria 

 of Newton, were shown to be of some value in facilitating in- 

 quiries into the character of the roots of an equation, as by 

 their aid the mere inspection of the coefficients would often 

 supply the information not otherwise to be obtained without 

 a lengthy and laborious analysis. The formulas referred to 



* Communicated by the Author. 



