224 On the Dispersion of the Optic Axes, and 



with propriety be called the potential, since the motion of the 

 system is potentially, or virtually, included in it is a function 

 of the second degree, composed of the three quantities X, Y, Z, 

 which are connected with the/ displacements , rj, , by the fol- 

 lowing relations : 



= _ _ - 



dz dy* dx dz 9 dy dx 



To show this, I make use simply of the consideration that the 

 motion must be such as to satisfy the condition 



T+T+f--> 

 dx dy dz 



which seems to be characteristic of the vibrations of light. But 

 the same condition allows us to suppose that the potential con- 

 tains not only the quantities X, F, Z, but their differential co- 

 efficients of any order with respect to the co-ordinates. This 

 supposition, however, is too general, and requires to be limited 

 by other considerations. Now the most natural restriction 

 which can be imposed consists in the assumption that the quanti- 

 ties of all orders are formed on the same type, those of any 

 order being derived from the preceding in the same way that 

 the quantities X, Y, Z are derived from , 17, : there are par- 

 ticular reasons also which go to strengthen this hypothesis, and 

 have led me to adopt it. Putting therefore 



d_Y_dZ V _<^_Z 7 _^_ 

 Al ~ dz dy 9 * " dx dz 9 ' ~ dy dx '* 



dYi d& v _ dZi dX l 7 _ dX, dT\ 



^~~dz~~dy 9 **~ dx dz' ^-dy"dx> 



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

 degree, composed of all the quantities X, Y, Z, Xi, Pi, Z^ X 3t 

 P 2 , Z-i, &c. ; and this is the " mathematical hypothesis" alluded 

 to in the beginning of this article. The hypothesis occcurred to 

 me more than three years ago (June, 1839), but I did not ven- 



