-~I 
Motion of Solid and Fluid Bodies. 17 
du _ du _ du _, 
(ea iia ae ae 
The equations of condition to be satisfied at the limits become, therefore, 
a — a 0 oe S05 0 (40) 
i. e. the three quantities }, 5, 9, retain their values in passing from one medium 
to another. In order to obtain a more geometrical idea of the nature of these 
equations of condition, in the particular case of plane waves, we shall assume 
d* aq ar d? d- ah 
See Sle I= [3 po Dm = 28, = Fp 
dt dt dt dt dt dt 
where (p, 7, 7) are the components of a transversal 7, and (/; g, 1) of another 
transversal 7,,, so that we have 
P=TCOEr, (P=Tcosy, m= 7 cosy, 
SS, CORN = Ty CoN h'= F cosy 
These quantities (7, 7,,) may be called the first and second ¢ransversals of the 
real molecular vibration which takes place along (a, 8, 7), and is equal to 7; and 
they have a striking analogy to the line that Professor M‘Cullagh calls a secon- 
dary transversal in his theory of light; but there are two such transversals in 
the theory of elastic solids, while there is only one in the theory of light. It 
should be remembered that the equations of condition at the limits (32-40) are 
perfectly general, and independent of the particular integral for plane-waves ; 
but the geometrical constructions, which I proceed to investigate, belong only to 
the particular integral. Previously to determining the meaning of equations (40) 
it will be useful to establish a theorem relating to surfaces of the second order. 
Let the surface be 
Ag + By? + cz? + 2pyz 4 2Erz 4+ 2rry = 1, 
then we shall have 
cos(p,7" 
pr = acosacosa’ + BcosBcosp’ + ccosy cosy’ + D(cosfcosy + cosy cos’ ) 
ates) + £(cosycosa’+ cosacosy’ ) + F(cosacosp’ + cosBcosa’), (41) 
pr’ 
where (p, p’, 7,7’) are the perpendiculars let fall on any two tangent planes and 
232 
