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LXXXYI. On the Transformation of the Equation of Motion 

 of the Dynamics of Continuous Media in the Restricted 

 Principle of Relativity. By H. T. Flint, M.Sc, Lecturer 

 in Physics, King's College, London*. 



THIS paper develops no new result, but it is hoped that 

 its appearance is justified by the compact and simple 

 expression it gives to some well-known ones. 



The notation should appeal to physicists, for it is the 

 natural extension to four dimensions of Gibbs's Vector 

 Analysis, which is employed in modern treatises on 

 Mathematical Physics, and has been found specially con- 

 venient in the theory of Electromagnetism. 



The notation is explained in a paper by Lewis and Wilson 

 (' American Academy of Sciences,' 1911), in the ' Quarterly 

 journal for Mathematics ' (vol. xlviii. No. 1), and also in the 

 'Philosophical Magazine' (March 1921) f. In the last of these 

 the writer used an extension of Hamilton's notation, but the 

 extension of that of Gibbs appears to be more convenient, and 

 is doubtless to be more generally adopted in the future* 

 We shall therefore express our results in this notation. 



One of the advantages of this notation is that its analogy 

 with that of three-dimensional vectors is very close, and this 

 is an advantage in dealing with four dimensions. 



Writers on this branch of dynamics always point out the 

 comparison, so that there is no need to go into it in any 

 detail %. 



By a linear vector function of a vector, r, is meant the- 

 expression : 



</>.r = AiAs.r + BiBs.r + dCs.r, 



in which the A's, B's, and C's denote vectors. 

 (j>. is a linear vector operator, and is denoted by 



<£. = A1A2. + B1B2. + C1C2., . . . . (1) 



the dot denoting that the operand is to multiply A 2 , B 2 , 

 and C2 scalarly. 



<f>.v is itself a vector, with three components parallel to 

 A b Bi, and C 3 . 



* Communicated by the Author. 



t See also a paper by Prof. McAulay, " Multenions," Proc. Royal Soc. 

 ser. A. vol. xcix. p. 292. 



J Laue, Ann. der Phys. Band xxxv. (1911) ; ' The Principle of 

 Relativity/ Cunningham (1914). 



