30 
Proceedings of Eoyal Society of Edinhurgh. [dec. 5, 
where V denotes the velocity of either of two simple waves, whose 
wave-front is perpendicular to (A, /x, v). Consider the case of wave- 
front perpendicular to one of the three principal planes ; {yz) for 
instance ; we have \ = 0 ', and, to make { } of (29) a maximum or 
a minimum, we see by symmetry that we must either have 
perpendicular to principal plane) 1=1^ m— o,n—o 
or (vibration in principal plane) l=o, m——v, n=ix 
Hence, for the two cases, we have respectively 
YihxdXiou perpendicular to yz . . V2=M + N-f(B-M)fi2 + (C_N’)j;2 . . (35); 
Vibration m 2 / 2 : V2=L-i-B/x2+Cj/2 + 4(H-l-I-L)|uV . . (36). 
19. According to rresnel’s theory (35) must he constant, and 
the last term of (36) must vanish. These and the corresponding 
conclusions relatively to the other two principal planes are satisfied 
if, and require that, 
A-L = B-M = C-H (37), 
and H4-I = L; l4-G = M; G-bH = H . . (38). 
Transposing M and N in the last of equations (37), substituting 
for them their values by (23), and dividing each member by ^y, 
we find 
~ ^ B — A ^ ^ ^ ^ 
/3y-a/3 ya-/3y 
whence (sum of numerators divided by sum of denominators), 
B-C _ C-A _ A-E 
ya — a/3 a(3 — /3y (3y — ya 
. . (40). 
The first of these equations is equivalent to the first of (37) j and 
thus we see that the two equations (37) are equivalent to one only ; 
and (39) is a convenient form Of this one. By it, as put symmetri- 
cally in (40), and by bringing (14) into account, w^e find, with k 
taken to denote a coefficient which may be any function of (a, yS, y) : 
A = A:(S-^y); B = ^(S-ya); C = KS-a/5)l ^ 
where S = ^{/3y + ya + a/3) f 
and using this result in (23), we find 
L=^[a(g+y)-S]; M=^-[d(y+«)-S]; N=k[y{a+g}-B]) 
or L=k(2S-/3y); M=/c{23-ya); N=/c{2S-ay) i ‘ ‘ ^ ’ 
By (2) we may put (41) and (42) into forms more convenient for 
some purposes as follows : — 
