96 Prof. Miller on the Anharmonic Ratio of Radii 



however, certaiu at present whether the metallic acid of Fergu- 

 sonite is columbic acid or tantalic acid. Hartwall did not find 

 alumina present ; and the mode of analysis adopted by him is 

 such that it does not differ from what is at present the usual 

 method employed in analysing such compounds ; and with the 

 exception that oxide of lanthanium is included in his oxide of 

 cerium, I doubt if M. Kenngott could suggest any cause for its 

 producing inaccurate results. 



In conclusion, although there can be no doubt that the mi- 

 neral species Tj'rite and Fergusonite are closely allied, and may 

 possibly be even identical, still until analyses of Fergusonite 

 confirmatory of this view are brought forward, these species 

 should most decidedly remain separate ; and little benefit to mi- 

 neralogy can arise from attempting to wring conclusions out of 

 data which all must admit are at present insufficient to decide 

 the case. 



XV. On the Anharmonic Ratio of Radii normal to Four Faces of 

 a Crystal in one Zone ; and on the Change of the Axes of a 

 Crystal. By W. H. Miller, M.A., F.R.S., Professor of 

 Mineralogy in the University of Cambridge *. 



THE anharmonic I'atio of radii normal to four faces of a cry- 

 stal in one zone, in the shape in which it appears in my 

 ' Crystallography,' art. 25, as well as in the new edition of Phil- 

 lips's ' Mineralogy,' art. 22, is expressed by three different frac- 

 tions. The necessity for employing three different expressions 

 for the ratio, which greatly embarrasses the application of one of 

 the most useful propositions in crystallography, is due to the 

 circumstance that one or two of them take the form §, for par- 

 ticular values of the indices. By a slight change in the inves- 

 tigation, I have obtained the result in a single general form 

 (equation (3) below),' of which the three fractional expressions 

 above mentioned are particular cases. 



Using the notation proposed by Dr. Whewell in the Philoso- 

 phical Transactions for 1825, and adopted in the 'Crystallo- 

 graphy,' and the new edition of Phillips's ' Mineralogy,' let efg, 

 hkl, iiqr, uvw be the symbols of any four faces of a crystal in one 

 zone. Let P, Q, R, S be the extremities of radii of a sphere 

 drawn perpendicular to the four faces respectively, or the poles 

 of these faces. Let X, Y, Z be the extremities of radii of the 

 sphere parallel to the axes of the crystal ; a, b, c the parameters 

 of the crystal. Then 



* Communicated by the Author. 



