VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 37 



1, Problem defined. — If a quantity A z =f(z) be expressed by 



the equation 



J z = A + Bz* '"+ Cz q + Dz T + etc (1), 



in which let us suppose the constants A, B, C, etc., have any 



Unite value positive or negative including zero, and the indices 



p, q, and r, etc., are in ascending order of magnitude and positive, 1 



the integral 



fA z dz = V = z (A+ B s p + -JL-^+etc..".) (2) 



will represent an arvsa included between the curve (1), the axis z, 

 and the limiting ordinates, z x and z 2 say, provided A z represents 

 an ordinate : and similarly it will represent a volume should that 

 function denote the area of xy planes at right angles to the z axis. 

 The volume will of course be that included between the parallel 

 terminal planes, intersecting the axis at the limits of the variable, 

 and the surface formed by the boundary of one of these, considered 

 as generator, moving along the z axis at right angles thereto, and 

 changing its area in terms of the function. 



Since the origin of z and the linear scale of the unit by which 

 it is measured do not affect the degree, but merely alter the con- 

 stants of the above expressions, viz. (1) and (2), these may be 

 regarded as quite general in form. A may consequently be con- 

 ceived as the ordinate, or as the area, for 2 = 0, and V correspon- 

 ingly as the area or the volume for z=l ; provided that the limits 

 of the integral be and 1, and the constants be suitably determined. 

 Hence, subject to the restriction defined, unity may be substituted 

 throughout for the quantities z, z v , z q , etc., in (2). 



Let a, b, c, etc., represent any proper fractions in ascending 

 order of magnitude; and a, /3, y etc. any series of weight-coeffici- 

 ents 2 to be multiplied into the values of the function, for values 

 of z equal to those fractions ; and for brevity let the sum of the 



1 Negative indices give a series of hyperbolas if A z be regarded as an 

 ordinate, the asymptotes being the axis z and the ordinate for z = 0. We 

 consider only the positive indices, that is the parabolas. 



2 We shall call these ( weight-coefficients ' because they express the 

 relative importance of the sections. 



