Mr. W. H.Walenn on Division-Remainders in Arithmetic. 345 



sin (%'— %), indifferently whether its sign (and therewith the 

 resulting elliptic polarization) is positive or negative. Under 

 this assumption, the difference of brightness between the edge 

 and middle of an absorption-band becomes greater in propor- 

 tion as D is less and consequently the band is narrower. 



Moreover the preceding discussions have cleared up many 

 formerly obscure points ; and if the approximative formula (9), 

 with corresponding treatment, may be applied to the absorp- 

 tion as well as to the refraction of composite media, on the 

 other hand it can scarcely be any longer doubtful that the 

 phase-difference (x'~~X) ^ oes n0 ^ ^ e ^ ne " coefficient of 

 ellipticity," diminish towards the interior of the medium, but 

 much rather remains constant, so that in each successive 

 stratum an equal quantity of regular is converted into irregular 

 oscillatory motion. 



[To "be continued.] 



XLIII. On Division-Remainders in Arithmetic. 

 By W. H. Walenn, Mem. Phys. Soc* 



EEGARJDINGr the dividend, divisor, quotient, and remain- 

 der in an ordinary numerical division as algebraical 

 quantities, the operation may be put in the form of the equa- 

 tion 



a r 



b =C+ b 



r 



The expression c+risa function of r ; it is also a function of 



c and b, and r cannot be obtained from it without knowing c 



as well as b. If the division svmbolized by T be carried so far 



that c is either wholly decimal or partly integral and partly 

 decimal, r is a decimal less than unity. The following are 

 instances of this : — 



I. || =0 -409+^ ...r=0-051. 



bl bl 



II. 315 =9-545+ ~ ...r = 0-015. 

 oo do 



III. ^1-527 + ^f... ,=0-006942. 



7-854 7*854 



* Communicated by the Author, having been read "before the British 

 Association, Section A, September 9, 1876. 



