4 
coordinates Ax’, ly’, Az', that is to say 
’ , 
X X 
My; = My e (8) 
z Zz 
ence, 
(1—X) — (1 —cos®) sin? (1 — cos) sing cos $ — sin#sin g 
(1 —cos 9 ) sind cos? (1-2) — (1 — cos:$) cos?g sin & cosg| = © (9) 
— sin Ÿ sin $ sin # cos cos # — X 
which is satisfied when A= 1, for in this case the first two rows become 
indentical. 
Substituting A= 1 in (8) we obtain the following set of equations 
defining the invariant directions 
— (x — cos 9) sin?$ . x + (1— cos) cos? sing. y'— sinB sin $ .z' — o, 
(1 — cos#) sin # cos. x’ — (1—cos?)cos^$ .y' — sine cos $.z' — o, 
— sin # sin d. x’ + sin# cos. y' + (1—cos#).z' = o, 
These equations are simultaneously satisfied only if z' = o, in which case 
they give 
sing. x = cosg.y, 
or, 
, 
y = Xx tang 
In other words, the invariant direction falls in the plane of the 
picture and is at right angles to the direction of tilt (angle # from the y: 
axis measured in the anticlockwise direction). Furthermore, since A = 1, 
all the points on the line y' 2 x' tang shall not change the values of 
their coordinates. 
3) No other line has the property proved above. To see this, expand 
the determinant in (9) after writing the element in the bottom right hand 
corner in the form — (1—À) + (1 — cosá) | We obtain 
(1 —X)3 — à (1 — A)? (1 — cos e) * (1— 3) 1 (1 — cos &)? 
(1 + sin^9 cos^9 ) + sin? cos? | 
— (1 — cos 8) sin? ¢ { (1 — cos 8)? + sin? 4 | cos? $ 
— (x - A) (1 — cos e)? sin? 4 . cos?$ 
+ (1 — cos ®) sin cos | (1 — cos # )* + sin? «| 
sin $ cos $ + (1 — X) sin“ ® . sing