APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 129 



SOME APPLICATIONS and DEVELOPMENTS of the 

 PRISMOIDAL FORMULA. 



By G. H. Knibbs, f.r.a.s., 

 Lecturer in Surveying, University of Sydney. 



[Received Aug. 29. Read before the Royal Society o/N. S. Wales, Sep. 6, 1899.'] 



1. The prismoidal formula and the limit of its application. 



2. The prismoidal formula applied to solids with ruled surfaces 



3. Skew, warped, or ruled quadric surfaces. 



4. Volumes of warped-surface solids. 



5. Solids of trapezoidal section with two warped surfaces. 



6. Solids of quadrilateral section with plane surfaces. 



7. Solids of quadrilateral section with one warped surface. 



8. Solids of pentagonal section with two warped surfaces. 



9. Solids of heptagonal section with four or six warped surfaces. 



10. Approximate estimate of volume from profile of centre-line of m 



longitudinally contiguous solids. 



11. Solids whose longitudinal axes are curved. 



12. Solids whose longitudinal axes are tortuous curves. 



13. Solids of curved longitudinal section with circularly warped 



surfaces. 



14. Centre of gravity of various sections. 



15. Prismoidal formula applicable to circularly warped solids. 



16. Prismoidal formula not applicable with variable radius of 



curvature. 



17. Suggestions respecting the use of the preceding formulae. 



1. The prismoidal formula and the limit of its application. — 

 Consider the solid generated by parallel motion in the direction z r 

 of any figure in the plane xy, whose area is connected with its z 

 coordinate by the relation 



A z =f(x.y.z) = A + Bz+Cz' i + Dz* (1); 



i.e., a solid whose xy section is a cubic function of its z coordinate. 

 Since the origin of z does not affect the degree of the equation, but 

 alters only the values of the constants, we may analytically treat 

 A as the area of one of the terminal planes of the solid, and express 

 its volume as 



• I— Sept. G, 1899. 



