XIV. ABSTRACT OP PROCEEDINGS. 



will not affect the degree of the equation; any fractions of the 

 axis must however be raised to the powers of the original function 

 (1) in order to determine its value for z, plus the fraction. Let 



V= L— { aA a + pA h + etc. i (3) 



a + /3 etc. ( J 



where A & is the value of the original function for % x + a/(z 2 - z x ) 

 and so on ; then (3) may be written 



V=A + -^(aa p + pb*+...) + —(aa (i + fib* + ) + etc (4) 



Equating (4) with (1); remembering that z in the latter is to be 

 considered unity, and u being any index as p, g, r, etc., we have 



a(a u - _A_V/3(6 U - — LA+etc. =0 (5) 



u + 1/ u + 1/ x 



as the fundamental equation in determining the relations between 

 the indices, the position of the sections of the axis, the number of 

 sections to be taken, and the weight-coefficients of the section. 

 The divisions of the subject as treated in the paper are as follows : 



1. Problem defined. 



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



and weight-coefficients. 



3. Determination of the ratio of the witl 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- ~k—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 formulas with symmetrical sections. 



