362 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



And this is true for ail values of : 



•'-de — ^' 

 and 



y=J^MdQ (1.) 



If we imagine tlie plane C Q to slide along the axis A z (fig. 4.) without otherwise 

 altering its position, the elementary solid included between it and P C will retain the 

 same volume as it had before ; for the two planes P C and Q C may be divided into 

 the same number of similar elements, whose corresponding angles being joined, the 

 solid element included between them will be divided into as many pyramidal frusta, 

 the volume of each of which will remain unaltered by the supposed displacement of 

 C Q, since each such frustum may be imagined to be made up of two pyramids, the 

 base of each of which will remain the same after the displacement, and their bases 

 and vertices between the same parallels. Thus, then, the volume determined by the 

 above formula is that of the conchoidal solid under its most general form. 



To determine the Area of a Conchoidal Surface. 



Let U represent the whole area of the surface (fig. 3.), and A U the elementary area 

 intercepted between the positions P C and Q C of the generating curve, supposed to 

 revolve without otherwise altering its position on the axis. 



Take A S to represent the element P Q of the curve described by the extremity P 

 of the revolving axis P C of the generating curve. 



Imagine the generating curve to describe, without altering its dimensions, an angle 

 about the axis A z, such that the circular arc, described on this supposition by the 



A S 

 point P, may equal P Q or A S. This angle will be represented by -^• 



The generating curve remaining always similar to itself, its statical moment about 

 A s$ is a function of or of R. Let it be represented by N, and considered a func- 

 tion of R. The elementary surface which the curve C P will generate, on the 



A S 

 supposition just made, will then be represented by -tq- • N, according to the property 



of GULDINUS. 



A surface similarly generated by C Q will in like manner be represented by 



RTXB^^ + ^N)- 

 Now the dimensions of the actual element of the conchoidal surface lie between the 

 dimensions of these two imaginary surfaces. 



This will be seen if we conceive any number of planes passing through the axis 

 C D, at right angles to A ^, to intersect all three of the surfaces spoken of. The 

 intercepted parts will be strips of the three surfaces, all of the same length, but of 



