•62 G. H. KNIBBS. 



respectively. The ordinates are identical for p-0, 1, and oo for 

 all values of k/n, and since in (48) the coefficient k is multiplied 

 into the term 2/(p+l) as well as the two other terms, it is evident 

 that a series of terms of the type (48), satisfying any system of 

 values of p, will always satisfy also the special values 0, 1 and oo . 1 



It may be remarked in regard to the k/n curves, that if for 

 certain abscissae, p> 1, their ordinates be greater than those of the 

 2/(p+l) curve, then for sufficiently large values of p, they will 

 become equal to the corresponding ordinates of the latter, and 

 ultimately less than them. Thus the graphs make it obvious that 

 the differences of the ordinates of the 2/(p+l) curve and the 

 others vary differently with p, excepting, as already indicated for 

 p — 0, 1, and oo ; and therefore also that, by combining the proper 

 number of curves, with suitable changes in their parameters, any 

 number of given indices may be satisfied. It is moreover also 

 evident, both from algebraic considerations, and from the graphs, 

 that by properly determining the coefficients, at least as many 

 different indices may be satisfied, 1 included, as there are terms 

 in (48). We shall shew later that a larger number may be 

 satisfied. 



When n = 1, that is when there are only terminal sections, p = 

 and 1 only can be satisfied : we have already seen that when n — 2, 

 that is when there are two terminals and a middle section, the 

 values p — 0, 1, 2 and 3, may be satisfied, the coefficient of the 

 middle section being 4, The case is instructive : equation (48), 

 reduced, becomes 



2-^-1) 



H -2v-(p + l) K ' 



1 For solids p — represents a cylinder ; p = to 2 conoids, the meridian 

 curves of which are outwardly concave, at the limit p = 2 becoming a cone; 

 j? = 2 to oo represent conoids whose meridian curves are convex outwards. 

 At the limit p= oo , it cannot be said that a real solid is represented. 

 A z = Bz (X is a straight line for z = to 2 = 1, coinciding with the axis itself, 

 at 2 = 1 + dz it becomes an infinite plane at right angles to the axis. 



