THOMAS U. TAYLOR-PRISMOIDAL FORMULAE. 
41 
If s* and S'% are the mid-seetions of the cylindroid, we have 
S % =|(B+C)-iA, 
S'a-KB+A)—id. 
Si^+S'i4=B-f-^(C+A), 
S 14 — S'y s = f(C—A). 
The bases (A) and (C) may be regarded as the bases of the cylindroid 
generated by line CA whose mid-section is generated by point O. 
.-. Yol. CA=i(C+A+40), 
=KC+A+B). 
The volume generated by the diagonal CA is one-sixth of the three 
cylinders on B, C, and A. 
The volume generated by the parallelogram = 
~ (area generated by line PC+area generated by line BA). 
For volume cylindroid 
Volume associate cone 
1 
7 
A). 
.-. Volume generated by parallelogram 
=|[(B-A)+C]. 
7. If in Figs. 4 and 5, S is any point on the line BC situated so that 
BS-pBC=m 
and CS-t-BC=n, 
then from Elliott’s Extension of Holditcli’s theorem we have 
S=- 
m 
n 
, B—™_-A, 
m+n m-j-n ’ 
m+n 
where S, C, etc., indicate the area traced by the points S, C, etc. 
If m=n=^- 
S=*C+P—SA=S*; 
but from Newton’s formula we have 
M=|[B+C+4Si^]. (3) 
Eliminating S 14 , we have 
B+C 
M=- 
.1 a 
6 A. 
[See Scientiae Baccalaureus, February, 1891.) 
5. The volume of any prismatoid (cylindroid) is equal to a prism (cyl¬ 
inder) and a pyramid (cone), the altitude of each being equal to the altitude 
of the prismatoid (cylindroid ) and their bases being respectively the mid-sec¬ 
tions of the prismatoid (cylindroid) and the associate pyramid (cone). 
