1893, 



.] Displacement of a Rigid Body in Space by Rotations. 147 



other parts for different times at a fixed distance from the light. As 

 a result it was found that, on development, the deposit was greatest 

 when the exposure had been made at 2 ft. and diminished for each 

 successive distance. By applying the measures of the different 

 blacknesses obtained at the different distances to the curve obtained 

 by the measurement of the scale of exposures, it was found that the 

 exposure at 24 ft. ought to have been prolonged by 4'3 times to 

 give the same blackness as that at 2 ft., the other distances giving 

 intermediate results. If the law held good, the actual blackness of 

 deposit at 24 ft. would have been obtained had the same exposure 

 been given at about 50 ft. Other experiments are in progress, but 

 it seemed advisable, without waiting for their completion, to make 

 this addition to the paper, to show that the law fails both when short 

 exposures and also feeble intensities of light are in question. 



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

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

 Received May 19, 1893. 



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

 aware, has hitherto been done the conditions of the arbitrary dis- 

 placement 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 Rodrigues 

 in his classic memoir " Des lois geometriques qui regissent les de- 

 placements d'un systeme solide dans 1'espace ... ." 'Liouville,' 

 vol. 5, 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. 3j)5 396), 

 where it is shown that (p. 390), " Tout deplacement d'un systeme 

 solide peut etre represente d'une infinite de manieres par la succession 

 de deux rotations de ce systeme autour de deux axes fixes non con- 

 vergents. Le produit 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 conjugues, au produit du sinus de ,la 

 de mi-rotation du systeme autour de 1'axe central du deplacement, 

 multiplie par la demi-teranslation absolue du systeme." 



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

 " Tout deplacement . . . peut toujours provenir, d'une infinite de 

 manieres, de la succession de deux rotations autours de deux. axes non- 

 convergent s pourvu que le produit. . . ." 



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



i, 2 



