CONTINUOUS CALCULATING MACHINES. 
393 
are worthy of consideration. In the first place its symmetrical proportions and wide 
range of action, as shown by fig. 27, are due to the use of the general form of the 
sphere and roller mechanism, by which the axis of rotation of the sphere can be made 
to change across through the centre of the recording roller (B), so as to enable the 
relative motion of this roller and that of the driving one (A) to be reversed, so that 
positive and negative values may be obtained. 
Suppose the pointer to be on that portion of the curve to the right of the line L M, 
up which the driving roller is passing {i.e., really the axis of x )—then it has already 
been proved that the area is integrated by the motion of B. But when the pointer 
passes the intersection of the line L M with the curve the relative motion of A and B 
is reversed; this causes B to turn in the opposite direction (unless the widest point of 
the area is accidentally reached at this point). But the pointer is really now travelling 
round the opposite way on the curve to the left of L M, and the dial of B is therefore 
only cutting out or subtracting the negative portion of the curve. When the highest 
point is readied then the motion of the roller A will be reversed, and thus along the 
rest of the curve to the left (since the pointer is still to the left of L M) the motion of 
the index of B is positive, i.e., the area is continuously integrated. 
Thus as far as the integration of areas is concerned it is absolutely immaterial how 
the parallel ruler or rolling integrator is applied to the paper, or along what line the 
roller A runs, whether within or without the curve. 
Coming now to the moment of an area it is evident that the position of the 
line L M has everything to do with the result, but it is clear that mathematically 
there is an essential difference between this and the first case, for now a negative 
value of y does not as in the first case give a negative result—from the equation 
1 Sl=\^y 9/ dx 
and on examination it is found that the index of the wheel or roller C which records 
this result always moves in the opposite direction to the index of B when y is negative. 
Lastly the equation for moment of inertia 
lz=z^yHx 
shows that in this case where y is negative the value of I is recorded in the same way 
as the area. It will be seen that the mechanism always effects this result, without 
adjustment or correction by the very principle of its actiou, and adds the moment of 
inertia for both sides of the line. 
This peculiar relation between A, M and I, though evident upon a consideration of 
what these quantities mean in practical mechanics is thus clearly brought out. 
There is no reason (in theory) why the number of spheres should be limited to 
three or the frames kept parallel or perpendicular to each other. Thus it is possible 
to obtain the integration of 
