Ball — Contributions to the Theory of Screus. 669 



Take for iiibtaiice the coefficient of x^ dividecl by that of .?-, which 

 is easily seen to be 



\_ dr£ !_ (l-f _ 



Pi ' dai^ • . • ^^ • ^^^, , 



and wo learn that this expression will remain absolutely unaltered 

 provided that we only change from one set of co-reciprocals to another. 

 In this /is perfectly arbitrary. Let us take it for instance to be the 

 function H, or the square of the intensity, and we see that 



111111 



-+-+-+- + -+ - 

 2h Pz Ih Pi p-o P^ 



must be an absolute constant so long as we are only concerned with a 

 group of co-reciprocal screws. It is easily shown that this constant is 

 zero, and thus we have a theorem otherwise proved in Screios, p. 149, 

 and of which the present theory may be regarded as a generalization. 



It will be readily seen that numerous results can be obtained from 

 the different coefficients of the powers of x, the absolute term being 

 for instance the Hessian. The functions added to the emanants might 

 also be an arbitrary factor multiplied into R. Indeed if the discrimi- 

 nant were formed of the function 



it would be easy to show that the ratios of the coefficients must be in- 

 dependent of the screw of reference so long qis they were co-reciprocal, 

 and thus a multitude of functions of /would be obtained which retain 

 their form so long as the screws of reference are co-reciprocal. Wc 

 might even discard this last condition by writing for the factor multi- 

 plied by X the most general value of the pitch multiplied by the square 

 ' of the intensity. 



