344 



Dr. A. Schuster. 



[Jan. 27, 





Fraction. 



Calculated. 



Observed. 



Difference. 









: 



4,308*28 



43A8-4CL 







.*. •1.7 

 + 1/ 



fil 37-53 



K 

 O 



. ft 

 : 4 



4383-95 

 *±ooo vo 



4384-04 



+ uy 



5430-47 



D 



. ft 

 : 4 



4fi^4-fi9 



4fi^4-Q 



•4)00-* V 



• .9 



^984*27 



O 



• 7 



4529-37 



4^90-3^ 



•A9 

 — U^S 



5 140-20 



ft 



. ft 

 : 4 



4405-89 





_ -3Q 



fil 37-53 



ft 

 4 



8 



: 



5370-34 



^37A-fifi 



4- -39 



fi0fifi\38 



ft 

 4 



: 



5308-08 



5308-1 1 



4- -A3 



5383-99 



ft 

 4 



8 



: 



4710-09 



471 A-8A 

 1 / -LU ou 



_-1Q 



— 1 y 



5271 - 06 



ft 

 4 



a 



4fil2-18 



4fi1 9-08 



_ -1 A 



5984-fifi 



ft 

 4 



Q 



: v 



4654-74 





-u -9 



5447*57 



ft 

 4 



Q 



4237*00 



493fi-75 



— -95 



5372-1 fi 



ft 



4 



Q 



: y 



4178-35 . 



4178-98 



Ttl / O -iiO 



— ■<Y7 



5341 -85 



ft 

 4 



Q 



: y 



4154-77 



41 54-9fi 



4- -1 Q 



5328-90 



ft 

 4 







: y 



4144-70 



41 44- 3 A 



— '40 



fi003-91 



ft 

 4 



• ±u 



4202'74 



49A9-74, 



_I_ -AA 

 ± UU 



fi009'33 



8 



9 



5341 -fi3 



5341 -85 



4- -99 



fi-24 



8 



: 9 



4992-24 



4QQ1 -8Q 



— OO 



4fi33-47 

 woo ± / 



8 



9 



4118'64 



41 1 8-Q4 

 *iio yy* 



4- -3A 



fil 92-45 



9 



10 



5573-21 



5^73-35 



4. '1 4 



5603-40 



9 



10 



5043-06 



5042-71 



-•35 



4692-02 



9 



10 



4222-82 



4222-88 



+ •06 



4707-92 



9 



10 



4237-13 



4236-75 



-•38 



4787-23 



9 



10 



4308-51 



4308-45 



-•06 



5233 71 



9 



10 



4710-34 



4710-80 



+ •46 



Appendix. 



The problem which we have to solve may be stated as follows : — 

 Given a certain number of quantities distributed at random between 

 two fixed limits ; form the ratios between every pair of them, 

 and find the expectancy for the number of these ratios which shall 

 within certain small limits agree with a given fraction. In the first 

 place, we remark that without detriment to the generality of the 

 problem, we may assume the lower of the limits within which all the 

 quantities are lying to be unity ; for if it is not, we may by means of 

 a common multiplier to all quantities reduce it to unity. 



Let « be the given fraction with which all the ratios are to be com- 

 pared, and let A be the higher limit which none of the quantities 

 shall exceed. Assume at first A to be smaller than the square of the 

 reciprocal of a.. Divide the range A to 1 into two compartments ; 



the first from A to -, and the second from — to 1. Let there be t 



ot a. 



quantities which I shall call a lt a 2 , . . . in the first compartment ; and 

 let there be r quantities 6 l5 & 2 , . . . in the second compartment. ISone 

 of the quantities within one compartment can form amongst them- 

 selves ratios which shall be closely coincident with a. : say « + 



