24 Prof. Newman, On a twelve place Table, &c. [Dec. 4, 1876. 



(2) Peofessor F. W. Newman : A twelve place table of the ex- 

 ponential function. Communicated by Mr J. W. L. Glaisher. 



{Abstract) 



The table gives e'"" from x = 0*000 to a; = 5"400 at intervals of 

 O'OOl to twelve places of decimals. 



The mode of formation was as follows : the values of e'" for 

 integral values of a? were first calculated to 16 decimals ; this table 

 ends of itself when e~-^ does not affect the 16th decimal, viz. when 

 X = 38, Next the intervals were halved, so as to allow to a; the 

 form n + ^. The author then interpolated between x = n and 

 x = n + ^, so as to give to x the successive values w, w + 0*1, 

 n + 0'2, n + OS, n + 0'4, n + 0*5. Thus a second table was formed 

 with X increasing by 0"1 at each step from x = till e"'' = ; all to 

 16 decimals. 



The author next calculated, with 12 decimals only, the inter- 

 mediate intervals, so that x might increase by O'Ol at each step. 

 First the intervals were halved so as to give to x the form 

 n + -^n + 0"05, n being a digit : and then four new values of x 

 between each successive pair of values were filled in by interpola- 

 tion. A third table was thus completed, which ends at a; = 27*63. 

 At this time it was not the intention of the author to proceed 

 further, and though the work was performed to 13 decimals, the 

 last figure was not retained. 



Afterwards a fourth table was calculated, in which the increase 

 was only 0*001 at each step : this extends from a? = to ic = 5*400, 

 and forms the contents of the present communication to the society. 



The whole work was performed by the author's own hand, and 

 the oidy formula used at all in the calculations or interpolations 



^, ^ h h' h^ 



was ^ ' = l±l+i:2±lT2:3 + '^'-' 



and since this is rigidly accurate the interpolations never involved 

 small errors. It will be readily seen how by the subsidiary tables 

 every fifth value in the great table was completely verified. 



(3) Mr G. Chrtstal, B.A. : On the effect of alternating in- 

 duction currents on the galvanometer. 



Mr G. Chrystal and Mr Garnett exhibited before the Society 

 some experiments on the Galvanometry of alternating induction 

 currents. These experiments had been made by Mr Chrystal 

 during an investigation into the accuracy of Ohm's Law, and 

 will be found fully described in an article by him on Bi- and 

 Unilateral Galvanometer deflection, in the Philosophical Magazine 

 for December, 1876. 



Mr Chrystal gave a sketch of his explanation of the phe- 

 nomena, which he attributed to the influence of the alternating 

 currents on the magnetism of the Galvanometer needle. 



