﻿790 Dr. L. Silberstein on the 



The first value <r = evidently represents the case of a 

 uniform flow in which the stream-surface has the same level 

 throughout, and it is natural to conjecture that the con- 

 ditional value er = given above applies to the case of a 

 " standing wave w in the canal. 



When o = 3(?, or) =4\(f) =4 . u 2 j2<jh. 



M becomes Xd^u^l -. . (6) 



Jo 



Thus, if a standing wave is to be maintained in such a 

 canal the stream velocity at a great distance from the 

 elevation (compared with the wave dimensions), above and 

 below it must be such a function of the depth as to satisfy 

 (0) identically. 



When the motion is irrotational (6) gives U 2 =gh, which 

 is the ordinary expression for long waves in such a canal. 

 The method is, however, inadequate for determining whether 

 the wave is to be produced in an unimpeded canal or by 

 obstacles placed perpendicularly across the stream. 



It may be noted that the type of motion given by u=XJz\h 

 cannot satisfy (0), while, on the other hand, the motion 

 represented by w=U(r/A)i can satisfy (0), provided the 

 surface velocity have the value TJ 2 = 2gh. 



LXXVI. Quaternionic Form of Relativity. By L. Silbek- 

 STEIN, Ph.D., University Lecturer in Natural Philosophy, 

 Rome*. 



IT has been remarked by Cayley t, as early as in 1854, 

 that the rotations in a four-dimensional space may be 

 effected by means of a pair of quaternions applied, one as 

 a prefactor and the other as a postfactor, to the quaternion 



* Communicated by Dr. G. F. C. Searle, F.R.S. 



t A. Cayley, Phil. Mag. vol. vii. (1854), and Journ.f. reine u. angeiu. 

 Mathem.vol.50 (1855) ; or 'Papers,' vol. ii. Cayley limited himself to the 

 elliptic, i. e. real, rotations, but the extension to the hyperbolic and para- 

 bolic cases was an obvious matter. For the whole subsequent literature 

 of the subject, see the article of E. Study in the EncyclopMie d. 8c, 

 Math., tome i. vol. i. fascicule 3, p. 452 ; Paris and Leipzig, 1908. See 

 also F. Klein and A. Sommerf eld's work XJeber d. Theorie des Kreisets, 

 iv. pp. 939-943 ; Leipzig, 1910. It was in fact a general hint at 

 Relativity' made by these authors on p. 942 that, after I had a whole 

 year tried in vain a great variety of quateraionic operations for 

 relativistic purposes, suggested to me the choice of the particular 

 form (1). 



