Propagation of Waves in a Resisted Fluid, 523 



upon each elementary slice of the asthereal fluid be linear 

 functions of^ and its differential coefficients. Hence the re- 

 sistance must be a function of the form 



^'Tt^^'U^-^^^T?-^ ^^" 



where Cj, Cg, Cg, &c. are certain constants depending upon 

 the constitution of the aether, and of the transparent substance 

 in which it is. 



Furthermore, we may assume that the slices vibrate accord- 

 ing to the cycloidal law, which gives 



dt^- "" ^' 

 Hence the expression for the resistance becomes 



(C, - C3 «2+ Cs «4 ... ) ^ + (C,- C4 nH C« 7*4 ... ) ^, 



where, for brevity, we have put 



Ci - C3 «2 + c, w* ... = p, C2 - C4 w2 + Cg «-* ... = s'. 



In exactly the same way we may show that the resistance 

 parallel to the axis of ^ is 



dri d^ri _, 



Now T = X-i^ + Y-l^. 



V dt V dt 



V dt V dt 



Hence we find that 



v\\dt) ^ \dt/ J^ v\dt^ dt"^ dt^dtj* 

 ^ _ q_(d^d_^ _dnd_ri\ 

 vKdt^dt dt^dtj* 

 which forraulce are evidently equivalent to 



dv^ 



(3.) 



T=.pv + q^^ 



v^ 



N=<7 — 



p 



where p is the radius of curvature of the curve of vibration, 

 whatever it be. 



Suppose now that the curve of vibration is a circle, or in 



2 M2 



