112 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
the angle between them being 6 ; let them cut cr in the straight lines and pq, 
and let IT cut V in the area IMPQN, hounded by the ordinates MP and NQ. Then 
^9(2 = NQ MP ; and therefore, by § 5, the moment of the area MPQN about OH 
is equal to a~. pq. Hence, by Guldinus’ theorem, the portion of V included between 
IT and H' is equal to cr .pg , d = a-jh X area pq qp . By summation, we see that 
V = d'\h X area cr. 
The cylinder, with the curve o-, may be supposed to be unwrapped on a plane. 
Hence when we are given the central section of the solid, and a plan showing the 
form of S and its position with regard to O, we are able to construct, by geometrical 
methods, a curve whose area will give us the volume V. Take a standard 
line OX on the plan. Through 0 draw a line inclined to OX at an angle wdiose 
circular measure is a, and let this line cut 2 in points M and N. Take abscissae 
OM and ON along the base of the given central section, and draw the ordinates MP 
and NQ. On a line O'X' take O'L' = 6a, and draw an ordinate L'gp such that 
L'p = MP, \j'q = NQ. The ditferent points p and q corresponding to different 
values of a will form a curve, whose area can be measured ; and this area, multiplied 
by a-jh, is the volume required.* 
If the curve 2 encloses the base of the principal ordinate OH, the continuity of the 
boundary of cr will be broken when the cylinder is unwrapped. The locus of the 
points p is then the top of the rectangle representing the complete cylinder, and the 
area to be taken is the area between this, the sides of the rectangle, and the curve 
which is the locus of q. Similarly, if any portion of the boundary of 2 is at infinity, 
the corresponding part of the boundary of a will lie along the base of the rectangle 
representing the complete cylinder. 
The area cr is unaltered by projecting it at right angles to O'X' in the ratio 1 : X, and 
parallel to O'X' in the ratio X : 1. Thus we shall have L'p = X . MP, Uq = X . NQ, 
the point L' being taken so that O'L' = 6a/X. When the solid is the standard solid, 
it is convenient to take h — a (= I), and X —• 277 ; the unwrapped cylinder then 
becomes a square whose base is unity and height unity ; and the values of Up and 
Uq are given by the third column of Table I. (p. 153). 
If, for example, we divide the standard solid into twenty equal portions by 
nineteen parallel vertical planes, and if the cylinder is supposed to be divided along 
one of the lines in which it is cut by the central plane, and then unwrapped, and 
projected vertically in the latio of 1 :277 and horizontally in the ratio of 277 : 1, we 
* Generally, let V be a portion cut out of a solid of revolution b}' a closed cylinder K, whose 
generating lines are parallel to the axis of revolution. Let F denote the section of the solid by a plane 
through the axis of revolution ; and let S be a curve lying in the plane of F and related to it in such a 
way that any ordinate MP (drawn to S from a base at right angles to the axis of revolution) is propor¬ 
tional to the moment, about the axis, of that portion of F which lies beyond MP. Then, if F is given 
geometrically, and if the section of the cylinder K and its j^osition with regard to the axis are given, we 
can construct a figure whose area will be proportional to the volume V. 
