173 
of EdMurgh, Session 1882 - 83 . 
posed of an endless succession of equal waves, disposed alternately 
on either side of a straight line. It belongs, then, to the class of 
transcendental curves typified by the curve of sines. 
Instead of the string we may put two obstacles, one at each end, 
against which the spring may press. It is obvious that the arrange- 
ment ABODE would be one of unstable equilibrium, so that some 
slight guide would need to be placed at C, in order to prevent the 
spring from flying to the one or to the other side. 
When the bending is slight, as shown in the first figure (Plate I.'^), 
the form bears a considerable resemblance to that of the curve of 
sines when flattened to the same degree ; but when the flexure is con- 
siderable, as in the second figure, the deviation from that form becomes 
marked ; the angle of crossing at the points A, C, E, necessarily is 
increased. The third figure shows the form of the spring, with its 
continuation, when the ends are so drawn together, as that the 
curve crosses the axis squarely. 
In the fourth figure the spring is shown as so much bent that 
the angle of crossing is obtuse ; the distance between the ends is 
less than the breadth of the loop at its widest part. In the actual 
figure this distance is less than half of the breadth, and hence the 
continuations of the loops intersect each other. 
When the ends are brought still nearer the loops come to cross 
each other more frequently. 
The fifth figure shows the form of the spring when the two ends 
are brought together. In this case the continuations of the form 
are all included in its counterpart on the other side and in itself. 
When the ends of the spring are crossed over each other, it takes 
the form of what is technically called a Mnk, as shown in the sixth 
figure. There the distance of the ends is less than half the width 
of the loop, and the continuations intersect each other. If that 
distance were augmented the loops would stand detached. 
In these changes we have a noteworthy instance of the danger of 
abstract reasoning as to limits. While the point C is being brought 
nearer to A, the number of undulations of the curve within the 
limits of the sheet increases, so much so, that if C were very close 
to A, the paper would be covered by a multitude of lines, which, 
however fine they may be, would tend to produce blackness. 
If, after the spring has been crossed, we allow C to come back to 
