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Mathematics. — ‘A formula for the volume of the prismoid.” 
By Prof. JAN DE VRIES. 
As is known the volume of the prismoid is given by the expres- 
sion 
I= (P+ Q+4M), 
where A denotes the distance of the parallel faces, P and Q repre- 
sent the areas of these faces and M stands for the area of the sec- 
tion with the plane bisecting the distance of the parallel faces. 
We wish to show here, that this formula forms a special case of 
p 
a more general one, in which MZ has been replaced by the area D, 
of the section with the plane dividing the distance h between P and 
Q in the ratio of p to g. 
By joining any point O of the face P with all the vertices of the 
prismoid and combining these lines two by two by the necessary 
planes the prismoid is divided into: 1° a pyramid with vertex O 
and base Q, 2°. a number of tetrahedrons with three vertices in P 
and one vertex in Q, 3°. a number of tetrahedrons with two ver- 
tices in each of the faces P, Q. 
Let us first consider a tetrahedron P, Ps Qs Q, of the last group 
and suppose that the edges P; Q3, Pi Qu P Qs, Po Q meet any 
plane parallel to FP; Pz and QQ, in the points Djs, Dias Dazs Dog. 
Then we have 
Dd 
Now this tetrahedron can be considered as the second of three 
tetrahedrons, into which a certain prism with three side-faces is 
divided by two diagonal planes. Of any of the two triangles iu the 
parallel planes limiting this prism two of the three sides are paral- 
lel to sides of the parallelogram Dig Das D4 Dia; 80 the areas of these 
figures are in the ratio of P,P, X Q3 Q to 2 times D}3 Dog XK Dis Pig, 
i.e. as (p + 4)? to 2pg. So the volume of the tetrahedron is equal to 
Pp 
Diz Dag = Dia Du = aap P, Py and Dis Du = Dag Dog = Qs QW 
1 2 
Boe esas 
3 2 pg 
? 
D' representing the area of the parallelogram. 
The pyramid with vertex O and base Q determines in the plane 
