170 The Rey. S. Hauenton on the Equilibrium and 
where 
'=al +nm + mn? + 2amn + 2a,ln + 2a,lm, 
Pp 
Q =m tin? +n? + 28,mn + 28,ln + 28,dn, 
Rr =cn? +2 + um? + 2y,mn + 2y,ln + 2y,ln, (29) 
KF sal + Bm? + yw + 2umn + 2y,ln + 28,/m, 
Go sal’ + pm + yn? + 2y,mn-+ 2uln + 2am, 
nw’ al’ + By? + yn? + 28,mn + 2aln + 2nlm. 
Now, it is a known property of surfaces of the second order, that if the equation 
of a surface be 
Pa? + aly? + R’2? + 2e'yz + 2a’az + Qn'ey = 1, (30) 
that the equations (28) will determine the directions and magnitudes of the axes 
of the surface :* the directions being (a, 8, y), which have three sets of values ; 
and also, if (a, 6, ¢) denote the lengths of the axes, they are determined by the 
equation, 
pee beg 
litle (er 
ev” having three values corresponding to the three sets of values of (a, B, y). 
Hence we may deduce a geometrical construction for the direction of vibration 
ay = 
of molecules corresponding to a given direction of wave-plane. 
Construct the six fixed ellipsoids : 
p=av+tny +z + 2ayz + 2a,c2 + 2aay = 1, 
ga By + 12+ Na” + 2B6,yz + 28,12 + 28,0y = 1, 
R = cz? + Ma? ry® + Wye + 22 + 2ysry = 1, (29) 
Foae+ Byte + 2Ly2 + 27.02 + 2B,ry = 1, 
G Sav? + Boy+ ye” + 2y.yz + 2az + 2ary = 1, 
Ho a,2’ + By?+ 72° + 2B,y2 +2aaz + 2nay = 1; 
and from their common centre draw the normal to the wave-plane, this will pierce 
the surfaces in six points; let the corresponding radii vectores be (p, p,,, p,,,» 
r,1,,»7),,)> With these construct the ellipsoid, 
12 2 a) pe - 
* Vid. Leroy, pp. 73, 156. 
