24 KEPORT — 1873. 



On Modular Eiiuallons. By Professor Henry J. Stephen Smith, F.M.S. 



On Triple Tangent Planes. By W. Spottiswoode, F.B.S. 



On the Calculation of Logarithms. By tlie Rev. Henry "VVacb, M.A., 

 Brasenose College, Oxford, Chaplain of LincoWs Inn. 



Fertile purpose of any further extension of oui- power of logarithmic computation, 

 the author thinks attention should be recalled to the principle of the method pro- 

 posed in 1845 by Mr. Weddle. An account of this method and of its history may 

 be found in Mr. Peter Gray's preface to his ' Tables for the formation of Logarithms 

 and Antilogarithms to twelve places,' published in 18G5. It combines with great 

 directness and simplicity the advantage of increasing in facility of application as 

 the number of places is increased to which the computation is carried. It may be 

 briefly described as a means for expressing all numbers, of whatever magnitude, in 

 terms of certain factors to any required degree of accuracy. These factors are of the 

 form l + 'l^.w, where wj is any integer and n any simple integer. When tabulated 

 they present the following series : — 



09,-999 

 981-998 

 97! -997, 

 '9G-990; 



95 -yo-T 



94 

 ■93 

 •92 



•994 

 ■993 

 •992 



•9l!-991i 



9099 

 9998! 

 9997 

 9996 

 9995| 

 9994I 

 9993! 

 99f)2' 

 99911 



■99999 

 ■99998 

 •99997 

 ■99990 

 ■9999-j 

 •99994 

 •99993 

 •99992 

 •99991 



&c. &c. 



l-ll-Olil-OOlll-OOOl 



1-2 1-02 l-002il^0002 



1^31^03!p003|1^0003 



1^4 1-04 1-004 1-0004 



1^5:l-0.5;l-005il-0005 &c. &c. 



l-6ll-061-006il-000lj 



l-7jl-07|l-007|l-0007 



1-8 1-081-008 1-0008 



1-9 1-09' 1^009l 1^0009 



For convenience the author proposes to call these the Constituent Factors, and 

 the former the negative, the latter the positive factors; and the tables of their 

 logarithms may be called positive and negative Constituent Tables. To find the 

 logarithms of numbers we use the negative table ; to find antilogarithms, the pos- 

 itive table. A single example will show how numbers may be expressed in terms 

 of the negative factors and of the integers up to 11. 



A number on which Borda and Delambre have operated, viz. 543839, working to 

 twelve places of decimals, may be taken as an example. DiA'ide by 10^ and 5, and 

 the number becomes 



1-087678. 



Om- next object is to desti'oy the significant figure 8 in the second place of decimals. 

 For this purpose multiply the number by 1 — ^08 or •92. This is the same thing as 

 to subtract from the number eight times itself advanced two places ; and the work 

 is as follows : — . 



1^0 8767|80;0 

 _870l|42[4 



"PO 00 06 37 6 



