228 
Proceedings of the Royal Society of Edinburgh. 
[Sess. 
or, say, 
U + mm + nri = 1 , 
(13) 
X, Y, and Z being respectively the cosines of the angles between the 
y and 0 , 0 and x, x and y, axes. Hence 
r 2 = p 2 [A(A -\-p/uiZ 4- qv Y) + pp(pp + qvlL + AZ) + qv(qv + AY +p/uX)] , 
or, say, as in (13), 
r 2 = p 2 ( AA' +p/x(py)' + qv(qv )') . 
Also, a, /3, y being the direction ratios of the magnetisation, the cosine 
of the angle between r and (a, /3, y) is 
cos 6 = -( ax' + fiy' + yz) — f (aA' + + y(qy)') — -(Xa +pp .ft' + qv. y) . 
Hence, with r 2 — xx' + yy' + zz\ applying the oblique-axes operator 
a I(4 + ^; + 4) 
we readily verify the results of § 5, now referred to oblique axes, 
A r = a cos 0 . 
A cos 0 — a/r . sin 2 # , 
A sin 0 = - a/r . sin 6 cos 0 . 
Therefore the former expressions for T' 2fJi and L'^ are directly applicable 
to oblique axes when the modified values of cos 0 and sin 0 are taken. 
The value of L' 0 becomes, since 2 . ( \/ul ) = 0, etc., as before, 
where 
L' 0 = M[A'(3a'2 - 1) + B'(3/3' 2 - 1) + C'(3y' 2 - 1)] 
P 
■ ( 14 ) 
A 
A 2 
[AA' +pfi(prf + qv(qv)'f 2 
c'=2. 
p‘ 2 fx 2 
[XX ' + Pfi(pfi)' + qv(qv)J 
q 2 v 2 
[AA' + P + qv(qv)'f 2 ’ 
Putting /3 = 0, y = 0, a — 1 we get 
«L' 0 = ^[2A' + B'(3Z 2 - 1) + C'(3Y 2 - 1)] ; 
MT 
so also ? L' 0 = n[A'(3Z 2 - 1) + 2B' + C'(3X 2 - 1)] , , . 
y L' 0 = ^[A'(3Y 2 - 1) + B'(3X 2 - 1) + 2C'] , 
( 15 ) 
instead of (8). 
