Mr. J. E. Napieb on Voting at Limited LiaUliiy Companies. 199 



of those more immediately concerned with the working of boilers. The 

 whole subject is yet to investigate, and science, economy, and common 

 humanity, all urge the necessity of begmning without delay. 



A paper was read " On a Method of Voting at Limited Liability 

 Companies more Uniform than the present, and its Expression by a 

 Mathematical Formula," by James K. Napier. 



On a Method of Voting at Limited Liability Companies more Uniform than 



the present, and its Expression by a Mathematical Formula, By James 



E. Napiee. 

 In the Joint-Stock Company's Act, 19 and 20 Victoria, cap. 47, the 

 thirty-eighth clause of Table B refers to the votes of shareholders as 

 follows : — 



" Every shareholder shall have one vote for every share up to ten ; 

 he shall have an additional vote for every five shares beyond the first 

 ten shares up to 100, and an additional vote for every ten shares held 

 by him beyond the fii'st 100 shares." 



I failed in my endeavours to discover any other reason for the adop- 

 tion of this scale of votes than that of limiting the influence of the larger 

 shareholders ; the limitation, however, has been done too abruptly. To 

 make the scale vary more uniformly is the object of the present paper. 

 The accompanying sketch shows the scale of the " Act," and the 

 proposed modifications. 



The number of shares is marked along a base line according to a 

 scale ; the corresponding number of votes according to the Act is 

 represented by the ordinates to the dotted lines ; and the proposed modi- 

 fications, by the ordinates to ihefuU lines. 



The shares are supposed to be divided into an equi-difierent series, 

 whose common difference is 4: thus, 4, 8, 12, 16, 20,24,28, &c., 

 shares ; and to each term of the series four votes are allotted. 

 The sum of any number of the terms, therefore, commencing at the 

 first, represents a number of shares, and the last term represents the 

 number of votes. Thus, in the example above, the third term 12 

 represents the number of votes corresponding to the sum of the three 

 terms, 4, 8, 12 ; or, in general, n being the number of terms, the 

 number of votes will be represented by 4 n, corresponding to the 

 number of shares represented by four times the sum of the series 

 (1 + 2 + 3 + 4, &c., n) = 2 n (n + 1). This expression will prob- 

 ably be considered by the shareholders of public companies as no 

 simplification of the present scale, but it can easily be reduced to plainer 

 English, and nearly to the same words as those of the Act referred 

 to, thus : — 



