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 : 



X= - Y=- Z=- 

 dz dy* dx dz' dy dx' 



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

 motion must be such as to satisfy the condition 



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 , ?, : there are par- 

 ticular reasons also which go to strengthen this hypothesis, and 

 have led me to adopt it. Putting therefore 



^^_ _dZ_ = dZ _dX _dX _ d_Y^ 



dz dy ' dx dz' dy dx ' 



. dY\ dZi _ dZ\ dXi _ dXi dYi 



dz dy ' 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, X^ Fj, Zi, X Z) 

 F 2 , Z 2 , &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- 



