296 Prof. MacCullagh on the Dispersion of Optic Axes, Sf-c. 



ding in the same way that the quantities X, Y, Z are derived 

 from £, >j, £; there are particular reasons also which go to 

 strengthen this hypothesis, and have led me to adopt it. 

 Putting therefore 



X-^_^? V _ rL? _ ^5 7 _</X dY 

 *~ dz dy> 1_ dx dz' x ~ dy dx' 



X 



rfY, dZ l __dZ l _dX 1 _dX l dY } 



2 d ,? rfz/' 2 d .r dz* 2 */ y d .r ' 



and so on, I suppose the potential to be a function of the se- 

 cond degree, composed of all the quantities X, Y, Z, X 1} Y 15 

 Z 1S X 2 , Y 2 , Z 2 , &c; and this is the "mathematical hypothe- 

 sis " alluded to in the beginning of this article. The hypo- 

 thesis occurred to me more than three years ago (June 1839), 

 but I did not venture to communicate it to the Academy until 

 the date of my second memoir (May 1841); and even then I 

 had not studied it with the attention which I now conceive it 

 merits. It was only very lately, in fact, in some conversations 

 which I had with M. Babinet during a short visit to Paris, 

 that my attention was strongly drawn to the subject of disper- 

 sion in crystals, particularly the dispersion of the axes of 

 elasticity. My thoughts then naturally reverted to the hy- 

 pothesis which I have mentioned, and since my return I have 

 found that it affords a complete explanation of all the phae- 

 nomena *. 



I have also found that it gives the general law, extended to 

 biaxal crystals, of that elliptic and circular polarization which 

 has hitherto been detected only in quartz and in certain 

 fluids; while for the case of rectilinear polarization it gives a 

 law (very possibly a true one) more general than that of 

 Fresnel, but quite as elegant, and differing very slightly from 

 it. The hypothesis, therefore, is still too general for our 

 present purpose. To make it include only those crystals to 

 which the law of Fresnel is rigorously applicable, the alter- 

 nate derivatives X 15 Y v Zj, X 3 , Y 3 , Z 3 , &c. must be supposed 

 to vanish in the function which represents the potential. 

 Then, the axes of coordinates having any fixed directions 

 within the crystal, the axes of elasticity will be the principal 

 axes of an ellipsoid represented by an equation of the form 



AxZ+ByZ + CzZ+ZDijz+ZExz + 2Fxy = 1, 



* I am indebted, for my information on the subject, to a short article, 

 drawn up by MM. Quetelet and Babinet, in the 'Bulletin of the Royal 

 Academy of Brussels, vol. ii. p. 150; as also to PcggendorfTs Annals, 

 vol.xxvi. p. 309 ; vol. xxxv. p. 81. 



