July 27, 1893] 



NATURE 



3'i 



deflection was adjusted by trial and error, so that when k was 

 pressed no further deflection took place. To secure this, at the 

 beginning of an experiment, the slider was placed so that when 

 k was momentarily pressed, the deflection of the electrometer 

 needle was increased impulsively. The amount of this im- 

 pulsive deflection was noted, and the slider moved so as to in- 

 crease the steady deflection nearly up to the point on the scale 

 reached by the impulsive one, and then another trial was made. 

 In this way, by watching the point reached by each impulsive 

 deflection, and then increasing the steady one almost up to that 

 point, the latter was increased until the former vanished — that 

 is, until the potential of the quadrants was that of polarisation. 

 The magnitude of this deflection was then noted and the polari- 

 sation calculated from it. 



All the results point to the polarisation being constant with 

 large electrodes, being independent of the strength of the solu- 

 tion and the intensity of the current. The results of one series 

 of experiments are given in the accompanying table. The varia- 

 tions in the figures do not occur in any order, and are all such 

 as might be expected in experimental results of this nature. 

 Some of the greatest variations were obtained in exactly similar 

 experiments performed at different times. 



Mean olarisation = 3-og volts. 



"On the Displacement of a Rigid Body in Space by Rota- 

 tions. Preliminary Note." By J. J. Walker, F. R. S. 



Having been led to study more particularly than, as far as I 

 am aware, has hitherto been done the conditions of the arbitrary 

 displacement of a rigid body in space by means of rotations only, 

 the results arrived at in the case of the single pairs of axes seem 

 to me of sufficient interest and completeness to warrant their 

 being recorded. 



A comparison of these results with those arrived at by Rod- 

 rigues in his classic memoir " Des lois geometriques qui r^gis- 

 sent les deplacements d'un sysleme solide dans I'espace . . . ." 

 Liouville, vol. v. 1840, at once suggesting itself, it may be 

 proper here to recall the substance of the latter, and show how 

 far they fall short of the object I propose to myself. The case 

 of displacement by successive rotations round a pair of axes is 

 discussed in § 13 (pp. 395-396), where it is shown that (p. 390), 

 " Tout deplacement d'un systeme solide peut Otre represent; 

 d'une infinite de manieres par la succession de deux rotations de 

 ce systeme aulour de deux axes fixes non convergents. Le pro- 

 duit des sinus de ces demi-rotations multiplies par le sinus de 

 Tangle de ces axes et par leur plus courte distance, est egal, 

 pour tous ces couples d'axes conjiigues, au produit du sinus de 

 la demi-rotation du systeme autour de I'axe central du deplace- 

 ment, miiltiplie par la demi-translation absolute du systeme." 



Then (p. 396) the converse of this theorem is affirmed, viz., 

 that " Tout deplacement . . . peut toujours provenir, d'une 

 infinite de mantins, de la succession de deux rotations autours de 

 deux axes nonconvirgents ^OMXvu <yie le produit. . . ." 



In this conversion of the theorem above, it is strangely over- 

 looked that a displacement is not defined by the direction of 

 axis, and amplitude, of the resultant rotation, together with the 

 magnitudeof the component of ih: coi responding translation along 

 that direction (for in this form the proof is given, the axis being 



NO. 1239, VOL. 48] 



drawn through one end of the common perpendicular to the par- 

 ticular couple in respect of which the theorem is demonstrated), 

 since these elements are common to an infinity of displacements. 



These being premised, the laws connecting pairs of axes by 

 successive rotations round which a given displacement of a rigid 

 body in space may be effected are as follows : — 



If the first axis is taken parallel to a aiven vector, (,", there 

 are four directions, to any one of which (f) its conjugate may 

 be parallel, viz., the sides common to two qu.adric cones, the 

 constants of which are functions of f and the vectors defining 

 the displacement. 



One of these cones, whatever the direction of i', passes 

 through the vector which i-; the axis of resultant rotation for 

 the origin, or, in other words, which is parallel to the central 

 axis for the given displacement. The other cone (K) passes 

 through a vector covariant with f, say fj. 



The direction {' and any selected one of the four vectors f 

 being taken for a pair of axes of rotation, the corresponding 

 amplitudes are thus determined, viz. that of the second rota- 

 lion is double the angle between the planes of the vectors f, 

 f and i, f,. And as, ^ being fixed, f lies on two cones, one 

 of which, K', contains a side (f,) corresponding to the side fi 

 of K, the angle of rotation round the first axis is double that 

 between the planes of the vectors f, C and (,", f',. The planes 

 of f, fi and C , C'l nieet in the vector parallel to the central 

 axis. 



The directions of the axes being fixed in accordance with the 

 above conditions, the locus of either axis is a plane, the places 

 of the axes in which are so related that the connector of the feet 

 of perpendiculars on them from any fixed point generates a 

 ruled quadric surface. 



As regards the reality of the conjugates (C) corresponding to 

 an arbitrary direction (f) assumed for the first axis, it may 

 suffice here to state that one real conjugate, at least, is insured 

 by taking as f any side of the quadric cone which is defined by 

 replacing fin the cone K with the vector parallel to the central 

 axis. The two cones, whose common sides are directions of 

 the corresponding conjugate, then both passing through that 

 vector, will meet in at least one other real side. 



Paris. 

 Academy of Sciences, July 17. — M. de Lacaze-Duthiers 

 in the chair. —On the discovery of the comet b 1893, by M. 

 F. Tisserand. — Expression of the resistance offered by each 

 ponderable molecule to the vibratory motion of the ambiant 

 ether, by M. J. Boussinesq. — On the generalisation of a 

 theorem of Euler relating to polyhedr.", by M. II. Poincare. — 

 Expel iments on the resistance of air and diverse gases to the 

 motion of bodies, by MM. L. Cailletet and E. Colardeau. 

 The experiments previously made on the resistance of air to the 

 motion of falling bodies, and performed at the Eiffel Tower, led 

 to varying results according to the pressure of the atmosphere. 

 In order to determine the influence of the pi-essure upon the 

 resistance, and also that of the nature of the gas, the apparatus 

 was enclosed in a cast-iron receiver of 300 litres capacity, into 

 which air or other gas could be pumped up to pressures ot 8 

 or 10 atmospheres. The apparatus consisted of a paddle-wheel 

 set in motion by a weight suspended by a string wound upon 

 the shaft. A double cock, with intermediate reservoir, per- 

 mitted the introduction of a known quantity of shot into the 

 cylindrical hollow of the driving weight, so as to increase 

 the weight without affecting the pressure. .\ key, worked from 

 the outside through a stuffing-box, enabled the experimenters 

 to replace the weight as often as desired without loss of 

 compressed gas. The downward motion of the weight became 

 uniform as soon as the resistance of the gas equalled the driving 

 weight. An electric contact inside the receiver connected with 

 a bell outside indicated the rate of rotation of the paddle- 

 wheel. The resistance opposed by any gas to the motion of a 

 plane was found to be proportional to its surface, the square of 

 its velocity, and the pressure and density of the gas. If two 

 planes are placed one behind the other at a distance equal to 

 their breadth, the total resistance is about i-i times that offered 

 to a single plane. Placing two planes o'i5 m. broad I m. 

 apart, the sum of their resistances did not come up to twice the 

 resistance of each. — Observations of the new comet Rordame, 

 made with the great equatorial of the Bordeaux Observatory, 

 by MM. G. Rayet and L. Picart. — On a relation which exists 

 between the formulae of Coulomb (magnetic), Laplace, and 

 Ampere, by M. E. H. Amagat. It is shown that W. Weber's 

 method of arriving at the values of the constants of Ampere's 



