of Edinburgh, Session 1885-86. 
513 
occurrence. When there are two complete sets of colours, the outer 
is always green or blue looking, and the inner yellow. 
The measurements of halos may serve to show the limits of error 
of the instrument. 
3. On a Model of the “ Half-Twist Surface.” By Professor 
Crum Brown. (Plate XVII.) 
The “ twist surfaces,” of which this is a case, stand in the same 
relation to the helicoid surface as the anchor-ring does to the 
cylinder. In the helicoid the generating line, at right angles to 
the axis, rotates about the axis as the point of intersection moves 
along it. In the twist surfaces the generating line is always at 
right angles to a fixed circle, and rotates about the tangent to the 
circle at the point of intersection, as the point of intersection moves 
round the circle. The species of twist surface is defined by the 
ratio of the angular motion of the generating line to that of the 
point of intersection. In the particular case illustrated by the 
model, the generating line turns through two right angles, while the 
point of intersection makes one whole revolution ; that is, the rate 
of angular motion of the generating line is one-half of that of the 
point of intersection. 
The idea of making such a model was derived from the “ one- 
sided surfaces” exhibited by Professor Tait, formed by gumming 
together the ends of a strip of paper, after giving it half a turn 
about its axis. Such a strip has only one side and only one edge, 
or, perhaps more accurately, its two sides are continuous, and its 
two edges are continuous. If such a strip is very narrow, and if it 
is so arranged that its central line is a circle, it may be considered 
as a portion of a “ half-twist” surface. Without entering into any 
detailed mathematical discussion of the surface, there are some 
points of interest which may be indicated. A straight line passing 
through the centre of the circle and at right angles to its plane, 
obviously lies wholly In the surface, as every generating line cuts 
it. We may call this line the axis of the surface. Every plane 
through this axis contains two generating lines ; the intersections 
of these pairs of generating lines lie in a straight line touching the 
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