41 



On Oscidatinff Spheroids ami Surfaces of Constant Action. 



15. To develope one of the properties of the principal foci and 

 principal rays of an optical system, we must introduce the considera- 

 tion of osculating spheroids, and surfaces of constant action. The 

 characteristic function V, the mode of dependence of which upon the 

 coordinates ^r, y, z, distinguishes any one system of rays from anj-^ 

 other, having the same kind of light and contained in the same me- 

 dium, is equal, as we have seen, to the definite integral ybds, that is 

 to the action of the light, taken from the luminous origin of the 

 S3'stem to the point x, y, z; the word action being used in the same 

 sense as in that known law, which is called the law of least action. 

 We may therefore give the name of surfaces of constant action, to that 

 series of surfaces for each of which the characteristic function V is 

 equal to some constant quantity, and which have for their differential 

 equation, ,. 



3K=0=^-^Jx+|j3,+ |3.. (U") 



In like manner, if we denote by F' the analogous characteristic 

 function of one of those focal systems considered in the eleventh 

 number, which have their light of the same kind and in the same 

 medium, but converging towards or diverging from one focus ; the 

 general expression of this function V will be V := v^ ■\- const., f 

 being the distance from the focus ; and the differential equation 



3.V{ = = 3F', (T") 



will represent a series of surfaces, which are analogous to the surfaces 

 (JJ"). In the case of ordinary light, these surfaces {V") are spheres, 

 and they may be called in geneml, spheroids of constant action ; the 

 VOL. XVI. a 



