26 RE PORT — ] 8 73 . 



After working to half the number of figures, we proceed by simple division ; and 

 the multipliers corresponding to the successive quotients are 



•999,999,7, -999,999,999, &c. 



This process may be regarded as a method of intei-polation, and it appears to the 

 author simpler and more direct than that of diflerences. It enables us, in short, by a 

 direct operation to express differences in terms of a limited number of known factors. 



The logarithms of these factors are determined with great facility from the fun- 

 damental series, 



log (l±!/) = ±y-iy'±-y'-i'/±S^c; 



iory being of the form •!'", this series converges with great rapidity as m increases, 

 so much so that for the latter half of tlie number of columns required in a consti- 

 tuent table only the first term of the series is required. Suppose, for instance, we 

 are working to twenty places, then the hyperbolic logarithm of 1 — -1" x 7 or of 



•99999,99999,3= --00000,00000,7. 



The determination of h'N'perbolic logarithms by this method is therefore peculiarly 

 easy, the logarithms of the last half of the factors being written down for inspection 

 without reference to the tables. 



A fuller development of this method, embodying perhaps some improvement in 

 its working, will be found in a paper contributed by the author to the ' Cambridge 

 Messenger of Mathematics,' which will appear in the September and October 

 Numbers of this year. The author has there furnished constituent tables for both 

 hyperbolic and denary logarithms to twenty figures ; and he has discussed the rela- 

 tion of the method to some modifications of it proposed by Mr. Gray and others. It 

 would occupy too much space to enter here on these collateral points ; but the author 

 doss not think any modification of the method hitherto proposed retains its elasticity. 

 It affords, at all events, a valuable means of calculating and testing isolated logarithms, 

 and of extending partial tables of logarithms, such as are given in Callet, to a high 

 number of figures. The principle, moreover, of reducing numbers to the form l-Q 

 .... or 1 00 ... . might be employed to facilitate the printimj; of tables of ten or 

 twelve figures. If the logarithms were tabulated of the integers up to 11 and 

 of the numbers between 1 and I'Ol or I'OOl, a short table of auxiliai-y constituent 

 factors would furnish the logarithms of aU other numbers by very simple calculations. 

 Such a plan would probably be an improvement on that of the partial ten-figure 

 tables published ten years ago by Pineto. 



Mechanics and Physics. 



On a Geometrical Solution of the followiwj problem : — A quiescent rigid bochj 

 possessing three degrees of freedom receives an impulse ; determine the in- 

 stantaneous screw about which the body commences to twist. By IIobekt 

 Stawell Ball, LL.D., F.B.S. 



I. 

 For an explanation of the language used, and for proof of several theorems, re- 

 ference must be made to " Theory of Screws," Transactions of the Royal Irish 

 Academy, vol. xxv. p. 157. 



All the screws about which the body can be twisted form a coordinate-system ; 

 one screw of the coordinate system can be found parallel to any given direction. 



An ellipsoid can be found such that the radius vector, from the centre to the 

 surface, is proportional to the twist velocity with which the body must twist 

 about the parallel screw, so that its kinetic energy shall be one unit. This is the 

 ellipsoid of equal kinetic eneripj. 



Let s be the screw about which an impulsive wrench, F,, constitutes the given 

 impulse. All the screws belonging to the coordinate-system which are reciprocal 



