The SPHEROID, in geometry, is generated by the entire revolution of a semi-ellipsis about its axis. When the revolution is made round the largest axis, the spheroid is called prolate; and when round the shortest, oblate. This last is the figure of the earth, and probably of all the planets.

To obtain the solid dimensions of a spheroid, multiply continually together the fixed axis, the square of the revolving axis, and the number 52359877, or 1 of 3-14159, and the last product will be the solidity; i. e. pttc the oblate, and pt cc the oblong spheroid, where p = 3.14159, t the transverse, and c the conjugate axis of the generating ellipsis. Or, multiply the area of the generating ellipse by of the revolving axis, and the product will be the content of the spheroid; i.e. t A= the oblate, and c A the oblong spheroid; where A is the area of the ellipse. E. g. Required the content of an oblate, and of an oblong spheroid, the axes being 50 and 30. Thus, 50 x 30 x 785398161178-09724 the area of the ellipse. And 1178·09724 × × 30=23561-9448

the oblong spheroid; and 1178-09724 × × 50 39269-908 the oblate one.

Dr. Hutton has demonstrated the rule above given in the following manner. Put ƒ BI the fixed semi-axis, IM the revolving

K

зemi-axis of the spheroid, a SI any semidiameter of the section N B M, b = IK its semi-conjugate, y = AE an ordinate to the diameter S I, or a semi-axis of the elliptic section A F C parallel to K L, and z = E F its other semi-axis, also r E I, s the sine of the angle AES, or of the angle K IS, to the radius 1, and p 3.14159. Then, by the property of the ellipse KS L,

cabbaa-xx bb ×

aa-xx a a

=yy;

Cor. 2.-If to or from pfrr, the value of the semi-spheroid be added or subtracted, pfrrx x the value of the ge

3 aa-xx

aaa

neral frustum K ACL, there will result ipfrrhh