168 Mr. N. Campbell on the 



may be compared with fig. 2 of the previous case ; here, 

 owing to smaller wave-lengths being associated with larger 

 velocities, we may observe the front part of the disturbance 

 alternating between advancing as a crest and as a trough. 



I£ we wish to regard the origin as the seat of a suddenly 

 created disturbance we must analyse y and y r into periods 

 and we find as before two types of integrals. 



Using certain known integrals involving Bessel functions *, 

 we may reduce the expressions to a form suitable for our 

 purpose : we find in this way 



I/o 



7T 



-=^{ a cos (,cVt)dfc+^r sin (^ ; t)dK\ .... (18) 

 JttJo Ztt Jo 



y r = ±( a cos {K(x-Yt)}dtc+ ~ ( e~ K ' x sin (x'Y't) d,c r , . (19) 



ZttJq 'ttJo 



where V = c sin 2 (\ica) ,/^xa ; V = c sinh 2 (j^tc' a) /j^tc' a. 



Comparing the expressions in (17), (18), and (19) we see 

 that they allow of the same interpretation as the similar forms 

 in the previous illustration. 



XIV. The Principles of Dynamics. By Norman Campbell, 

 Fellow of Trinity College, Cambridge], 



1. Introduction. 



2. Definition of the problems discussed. 



3. The relation of mathematics and physics. 



4. An example of this relation. 



5. The relation " A." 



6. 7, 8. The fundamental conceptions of dynamics. 

 9. The application of dynamics to experiment. 



10. The assumptions that must be made to make such application 



possible. 



11. "Absolute and relative motion.'' 



12. " The velocity of the sun in space." 



13. "Absolute translation and absolute rotation." 



§ 1. TT is doubtless rash nowadays to attempt to offer any 

 X remarks on a subject which has been so much 

 discussed as the basis of dynamics with the hope that they 

 shall be at the same time novel and useful. Ever since the 

 publication of the brilliant treatise of Prof. Mach on the 

 history of Mechanics the questions which he raised have been 

 canvassed eagerly by inquirers of every nation, and it might 



* Nielsen, Cylinderfunktionen, p. 195, 

 t Communicated by the Author. 



