372 Proceedings of Royal Society of Edinburgh. [sess. 
i) i\ i , j f Jr The value of y for each of the five waves is zero, 
and the values of a and (3 are as shown in the following table — 
a 
/3 
I 
+ sin i 
- cos i 
V 
+ sin i 
+ cos i 
I. 
+ sin i , 
- cos i, 
J' 
+ siny 
+ cos j 
+ sin 
- cos y, 

The section of the five waves by OX is the same for all, being 
expressed by ut / sin i for I, and by corresponding formulas for the 
four others. Hence if we denote r-times its reciprocal by a, 
we have 
sin i _ sin i t _ sin j _ sin j t / ^ x 
~ u u t v v t 
where u and u t are the propagational velocities of the distortional 
waves, and v, v 4 , those of the condensational waves in the two 
mediums. If now we take 
b = a cot i = J ( u ~ 2 - a 2 ) ; b t = a cot i t = J { u ~ 2 - a 2 ) ; 
c = a cot j = J ( v~ 2 - a 2 ) ; c=a coty^ = J ( v~ 2 - a 2 ) (18) , 
we have for the arguments of / in the five waves 
t-ax+by; t-ax-by; t - ax + b t y ; t-ax-cy ; t-ax-\-c t y (19). 
§ 10. Following Green* in calling the two sides of the inter- 
face the upper and lower medium respectively (and so shown in 
the diagram), we have for the components of the displacement in 
the upper medium 
£ = blf{t -ax + by) - blf(t -ax- by) + aJf(t -ax- cy) ) 
7] = alf(t - ax + by) + aYfit -ax- by) + cJ'/(£ - ax - cy) ) 
Green’s Math. Papers , p. 253. 
