36 G. H. KNIBBS. 



On the RELATION, in DETERMINING the VOLUMES 



OP SOLIDS, WHOSE PARALLEL TRANSVERSE SECTIONS ARE 

 fl ic FUNCTIONS OP THEIR POSITION ON^THE AXIS, BETWEEN THE 

 NUMBER, POSITION, AND COEFFICIENTS^ THE SECTIONS, AND THE 

 (POSITIVE) INDICES OF THE FUNCTIONS. 



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

 Lecturer in Surveying, University of Sydney. 



[Bead before the Royal Society of N. 8. Wales, June 6, 1900.'} 



1. Problem defined. 



2. General relation between indices, number and position of sections, 



and weight-coefficients. 



3. Determination of the ratio of the m + 1 weight-coefficients, when 



the number m of indices is one less than the number of 

 values of the variable. 



4. Number of indices greater than the number of values of the 



variable, diminished by unity. 



5. Number of indices less than the number of values of the variable* 



diminished by unity. 



6. Determination of the n-h = m weights. 



7. Position of a single section. 



8. Positions of two sections. 



9. Limiting positions of two symmetrically situated sections. 



10. Two symmetrically situated sections and their conjugate indices. 



11. Asymmetrical positions of two sections. 



12. Three symmetrical sections, viz., a middle and the terminal 



sections. 



13. A middle section, and two other sections equidistant therefrom, 



all of equal weight. 



14. Two terminal and one intermediate section. 



15. General result of the method of finite differences. 



16. General theory of symmetrically situated sections with sym- 



metrical weight-coefficients. 



17. Examples of the application of the general formula. 



18. The number of indices satisfied by a given number of symmetrical 



sections. 



19. Manifold infinity of possible formulae with symmetrical sections. 



