and Refraction of Solitary Plane Waves. 187 



orientation of equilibrium, and k as the bulk-modulus, that is 

 to say, the reciprocal of the compressibility, of ether. Thus 

 we now have as before in equations (15), (11), and (18) 



P • ^2. J^ — 1,-2— P 



a 2 + b 2 = u~ 2 =^; a 2 + b 2 = u { 



a 2+ c 2 = v -2 = P_. a 2 + c 2 = v -2-P 



(24). 



K Kj 



Using (20) and (21) in (23) with y = we nod 



n(a? + b 2 )(I + I') =71^ + 0?)!, J ' * [ ] ' 

 whence by (24) 



pJ'=p,J<; p(i+i')=M; (26). 



By these equations eliminating \ and J, from (22), we find 

 -{bp J -b J p)l+(bp l + b l p)V=a{p l --p)Z l 



X,,} ■ ■ ■ m- 



«0»-p)(I+I') = -(cp 



and solving these equations for 1' and J' in terms of I, w 

 have 



jr_ (bp-b^jcpt + cp) -a 2 (p-p) 2 j 

 {bp l + bfi) {cp, + c t p) + a- (p-p) 2 , 



j, = -2ab Pl {p-p) • * 



^P,+ b iP) {cpi + W) +a 2 (Pi-py 



and with J' and V thus determined, (26) give J y and I,, 

 completing the solution of our problem. 



§ 13. Using (18) to eliminate a, b, b n c, and c t , from (28), 

 and putting 



, P/ ~ P ,. =h (29); 



we find 



r _ p, cot i— p cot i-h(p-p) 



I /^coU + pcot^ + A^-p) ^ oU ^ 



and 



J /_ -2/^ cot? 



. . . (ol). 



I p, voti + p cot ij-\- h(p j— p) 



the case of t? and ^ y ve 

 ad Uj\ which by (2S) make 



cotj^l/va, and coty y = l/v y a . . . (32). 



Consider now the case of v and v, very small in compa- 

 rison with u and w y ; which by (28) makes 



