84 NOTE ON GULDIN’S PROPERTIES. 
employed his method to furnish .the demonstration of these theorems, 
which Guldin had not been able to obtain. The “centre of gravity” 
referred to in these theorems is the geometrical centre of mean posi- 
tion, as the theorems do not depend on any mechanical principles 
but are purely geometrical in their nature, and are simple applications 
of the integral ealculus. The usual statement of the first of these 
‘“‘Guldin’s Properties’ is as follows : ‘ 
‘If a plane area revolve about an axis in its own plane, the volume 
generated is equal to the product of the area and the length of the 
path of its centre of gravity.” 
_ This statement, however, ought to be limited to the cases where the 
area lies wholly on one side of the axis, as otherwise the product 
spoken of is equal not to the whole volume but to the difference of 
the volumes generated by the parts lying on opposite sides of the 
axis. ‘This extension is sometimes useful, as, for instance, in the in- 
_vestigation of the metacentre of a floating body. Here, if the dis- 
placement be made round the axis passing through the centre of 
gravity of the plane of floatation, it follows at once that the wedges 
generated on opposite sides of this axis are equal, and the whole 
.volume displaced therefore remains the same; or, conversely, if the 
displacement be made so that the volume displaced remains the same, 
and the wedges on either side of the line of displacement are therefore 
equa}, it follows that this line passes through the centre of gravity of 
the plane of floatation. This also gives at once the solution of a 
problem set in the Senate House, 1848: ‘‘A plane moves so as always 
to enclose between itself and a given surface S, a constant volume. 
Prove that the envelope of the system of such. planes is the same as 
the locus of the centres of gravity of the portions of the planes com- 
prised within S.”’ 
If we suppose the axis of revolution in the statement to remove to 
_an infinite distance, we have the case of a plane area moving parallel 
_to itself, while its centre of gravity moves in a straight line perpen- 
‘dicular to the plane of the area, and Guldin’s property holds not only 
swith regard to the centre of gravity but also to every point of the 
‘area. A similar extension applies to the following : 
; ‘If a plane area move parallel to itself, its centre of gravity moving 
in a curve, the plane of which is perpendicular to that of the area, 
is ‘the volume generated is equal to the product of the area and the 
‘projection of the path of the centre of gravity on a plane which is 
