VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 71 



equidistant from the centre, then if one of the sections be a middle 

 section the function satisfied will be of the same degree as the 

 number of sections, but if there be no middle section the degree will 

 be one less than the number of sections. 



19. 1 Manifold infinity of possible formulce with symmetrical 

 sections. — It has been shewn that for any given number of indices 

 a certain number of sections must be taken, these having definite 

 weight-coefficients. In symmetrical sections it has also been 

 demonstrated that if m be the number of different coefficients, 

 excluding that for the terminal sections, the degree of the function 

 satisfied will be 2m + 1. If k coefficients more than are necessary 

 in a particular case are taken, then a Mold infinity of formulae 

 may be developed. 2 



1 Added 16th June, 1900. 



2 The solids referred to in Note 1, page 62, are solids of revolution 

 merely. The form of the xy function is however quite immaterial. 



