196 The Rev. S. HavGurTon on the Equilibrium and 
_ (a & - 
BE =O) 
s=n(4H)  o=a(£+8, 0 
1=x(3 a 4) ==" 0! 
As the conditions at the limits express that the real vibration is preserved, 
together with the perpendicular component of the jirs¢ transversal and the hori- 
zontal component of the second transversal ; and as two of the four reflected and 
refracted vibrations are normal, and two transversal, it will be necessary, in order 
to express the conditions at the limits, to calculate the two transversals for both 
kinds of waves. 
Substituting, therefore, in equations (64), for ¢ and ¢, their values for a single 
wave, 
= 7Tcosacos@, ¢ = Tcosy cos, 
we obtain 
Qr Qicosa . 
qe = NT 810g, 
Q7r se tt de 2ncosy a 
=  F =_ SN 
2 Xr 2) 
2 lcos ncosa . 
= =| Nr te sing ; (67) 
and by a similar process from (66), remembering that now the vibrations are 
normal, 
2 20? 
— p= Nr. eee xX Sings 
2 1 2) 
= q = NT. sing, te = =(Nz: = sing. (68) 
Dn2 
ae = ep ais set sing, 
