206 M. Lorenz on the Theory of Light. 



in which 



_ a p x + b p y + c p z + d p 

 Pp- dp > "p + Op + Cp-*- 



The coefficients fl, e p} &c. which occur here are constant, and 2 

 denotes the sum for all values of the index p. 



Thus expressed, the function co or — ^ is retained with all its 



generality ; but if we preserve this degree of generality, which 

 comprehends every possible conglomeration of transparent sub- 

 stances, the integral can obviously represent only a confused 

 mixture of luminous motions. Without altering the form of 

 equation (1), we will therefore introduce an essential limitation, 

 by making the quantities d p very small. On this supposition 

 the formula will express a periodicity and a regularity such that 

 it will rapidly repeat itself at the various points of the body. 



We shall thus obtain a first approximation by taking the quan- 

 tities et p very small; this case is by no means the same as that 

 of perfect homogeneity, but directly leads, as we shall see, to 

 double refraction. 



The components £, 77, £ can be expressed by a series of the 

 following form : 



l = foC+2?(±/. p )C(±p p )+22f(± Pp ±p,)C(±p p ± / D,) + ..(2) 



In this expression f , f( -±p p ), &c. stand for constant coefficients 

 of the variable quantities C, C(+p p ), &c, which latter are ab- 

 breviations of the following values, 



cos (kt — Ix — my — nz) = C, 



cos ( kt — Ix — my — nz + p p ) = C ( + p p ) , 



&c. 



The double sign denotes the sum of the two expressions ; and 

 lastly, 5) and S% are the sum and double sum for all indices p 

 and q, both of which pass through the same series of values as 

 index p in equation (1). 



In the double sum, as well as in the subsequent terms, such 

 terms as have already occurred in previous terms must be con- 

 sidered as excluded ; thus, for example, the term 



i(p P -P<i)Q(Pp-Pq) 



would be excluded when q was = p. 



It will be understood that two quantities p p and p q , when p 

 and q are unequal, can be taken as different in such wise that 

 their sum or difference cannot become constant. On the con- 

 trary, it is conceivable that 3, 4, . . . &c. of the quantities p p may 



