EEPORT — 1888. 

 One of the forms into which De Foville throws this inequation is 

 (' Journal de . . Statistique/ 1886, p. 14). Another form (suggested in his 



226 



E< ~c, 



Sr 



E < ^c,. 

 &<■ 



paper dated 1879) is Cj + p.^.^ + p^c^ + &c. ; where the years (or short periods) are 



reckoned hackwards from the jjresent, and where p.,, pj, &c., are fi-actions 



corresponding to the greater loss whicli the coinages of earlj periods, as compared 



with the most recent, have undergone. 



Let Cj, e„ he the (unknown) numher of coins bearing each date in the existing 



circulation. By hypothesis 



Xi _ fij _ Cj _ ^>- {"W 



fc> s, s.. 



Also 



1 "2 

 E = Cj + e, + i&C. + Br 



e„ 



(2) 



Cr 



Co Cr 



Cj +Co- +&C. + Cr — 



C, + Co- + &C. + r- 

 "^ Cr 



Now each of the fractions of the form -i is reducihie to -i ; multiplying both the 



E 



numerator and denominator by ^ and taking account of equation (1). The 



o 



factors in this new shape are known quantities derivable from our data. Call them 

 1, p„, p3 . . . pr, respectively. Then we have 



E = -1 Ci + p.,e., + &c. + PrCr I 



Whence, as J^ is a proper fraction (or at any rate not greater than units), we have 



E< (or at most = ) C, + p.,c.y + «S:c. + prC^. 



A geometrical illustration may put the matter in a clearer light. In the 

 annexed diagram the line SS' represents the total number of samples, and its 

 segments «i, Sr,, &c., the number of samples bearing date 1, 2, 3, &c. The line 

 EE', divided into the proportionate segments e^ e., &c., represents the total number 

 of coins in circulation and the numbers thereof bearing each date. The numbers 

 of coins issued from the Mint each year are represented by the lines Cj c._, &c. In 

 general, the more recent the year, the less has the coinage of that year lost, the 



smaller is the ratio - . But, as we shall have occasion to remark afterwards, there 



may be an exception to this rule, as the figure shows in the case of the year 5. 



Now the essence of the reasoning is that EE'<CiC/ (and than any correspond- 

 ing line, e.g., CoC'^). Whether is it easier to say (a) with us, EE' <C - - Cj or (/3) after 



De Foville's second redaction (1885) EE'< SS' ^-\ or (y), in the spirit of his first 



paper, EE'<:Ci +«„ + a<j + &c., where a.,, a^, &c., are the remaining segments of the 

 line * CjC'j. Any of these segments a may thus be expressed in terms of Cr and 

 other known quantities : 



' ^, 183 &c. might be used to designate the ' remaining ' segments of Cj C\. 



