54 
TRANSACTIONS OR TIIR TEXAS ACADEMY OP SCIENCE. 
If in (42) we make S x =M=-|-(B-f-C-f4S ! 4) we get the value of x 
where the sections are equal to the mean area. It is thus possible to get 
a one-term formula. 
If Q=3B+C—4 Si / 2 , and R=2B+2C—4S*, we have 
S x =M=i(B+C+4Si /2 ) = B— Ox+Rx 2 , 
* or Rx 2 —Rx=i(C+4Si /2 —5B). 
By solving this and substituting the roots Xi and x 2 in (42) we get M. 
.-. V=MH. 
If in (32a) we make S X =M=^(B+C)—|A and make z=l —x we get 
S X =|(B+C)—|A=B+x(C—B—A)+Ax 2 . 
This equation will also give the values of x where the section S X =M. 
In the same way we can get a one-term formula from this associate as we 
did from (42). 
I find 
x 
2A 
(A+B 
— c ) ± [- 
3(B-C) 2 +A 2 
12 A 2 
BIBLIOGRAPHY. 
(1) Prismoid. 
Newton (Methodus Differentialis), 1711. 
McLaurin in a commentary, 1742. 
Koppe in Crelle’s Journal, 1838. 
Steiner in Crelle’s Journal. 1842. 
Koppe’s Nuer Lehrsatz, 1843. 
Ell wood Morris in Journal of Franklin Institute, 1843. 
Gillespie in Journal of Franklin Institute, 1857. 
Hermann Kinklin in Grunert’s Archiv, 1862. 
Mehler’s Elemental’ Mathematik, 1864. 
Zomlich’s Zeitschrift, 1878. 
Halsted’s Mensuration, 1881. 
Echols in Scientise Baccalaureus, 1890. 
Halsted in Scientise Baccalaureus, 1891. 
Echols in Annals of Mathematics, 1894. 
(2) EarthivorJc. 
Tables for Calculating Earthwork. By McNeill. 
Easy Rules for Measurement of Earthwork. By Ellwood Morris. 
Roads and Railroads. By W. M. Gillespie. 
Excavations and Embankments. By John Trautwine. 
Tables for Embankments. By J. R. Hudson. 
