of Edinhurgli, Session 1883-84. 
297 
to balance each other, or may happen to concur in one direction, 
hence there is a possibility of error in the last place, proportional to 
the complexity of the work. This error may, in the present 
instance, amount to 2 or even 3 units in the last place. Hence it 
follows that, in shortening from ten places to the six places in the 
final table, an uncertainty in the sixth place must occur when the 
rejected figures are within the limits 4997 and 5003, as happens 
thrice in the course of the work. Hence the necessity for founding 
our working tables on more extensive ones. 
All possible cases of the mechanical problem are comprised 
between the limits i = 0 and ^ = Jtt ; while i passes from the one to 
the other of these limits, the line AG makes the quarter of a turn, 
and we naturally inquire. What happens when the motion is farther 
continued ? or how was it on the other side of the zero ? 
On changing the sign of ^, the expressions for B - P and B - W 
change places ; the wheel takes the place of the pulley. This is 
clearly seen in the second figure ; 
the change from DC with its 
inclination inwards, to D'C' 
inclined outwards, is accom- 
panied by a reversion in the 
sizes of the circles. Hence it 
will suffice for us to examine 
the phases for values of i be- 
yond the right angle. 
In order to this, let the circle 
described with the radius AE, 
fig. 3, represent the pulley, 
which we shall suppose to re- 
main unchanged while the 
inclination of the band varies. 
When that inclination is zero, 
as in the' position D^Cq, the 
wheel is of the same size as the 
pulley; but when the band is Fig. 2. 
inclined, as in the position DC, the radius of the wheel has grown 
to be BD or BM, and the half length of the band is made up of the 
three parts ED, DC, CE. When i becomes a right angle, the points 
