60 G. H. KNIBBS. 



alteration of 2^0 in the coefficient 1 of z? We pass on therefore 

 to the consideration of the general theory of symmetrical and 

 symmetrically weighted sections. 



16. General theory of symmetrically situated sections with 

 symmetrical weight-coefficients. — The general theory of the relation 

 of indices, sectional positions, and weight-coefficients is sufficiently 

 indicated in § 2 - § 6 : it is proposed now to consider only the case 

 where both the weight-coefficients and the sections are symmetri- 

 cally disposed with reference to the middle-section, the first and 

 last being at the terminals of the axis. Equation (5) then takes 

 the following form, viz. 



MGH'.-.iJT-jW •■-«;; 



n being the number of parts into which the axis is divided, so 

 that including the terminals there are n + 1 sectional points. 

 When n is odd, k has the values 0, 1, 2,... J in - 1), that is there 

 are J (n + 1) terms : but when even, 0, 1, 2...^n, that is there are 

 ^ (n + 2) terms. It is important to remember in the latter case 

 that the final value of the weight-coefficient is one-half its proper 

 value ; that is the coefficient to be applied to the middle section 

 is double that in the formula : in other words if k be the final 

 weight-coefficient in the formula, 2k will be the proper weight- 

 coefficient. 



From (48) it is obvious, as we have before seen, that for p = 

 and p = 1, the equation is satisfied, whatever the values of k, k or w, 

 since in either case each term is zero, and the expression becomes 

 simply a0 + /30 + etc. = 0. Nevertheless, regarding each term as 

 a function of p, it is represented by a continuous curve, whose 

 abscissae are the values of p, and whose ordinates for p = + dp 

 and p = 1 + dp are perfectly definite. This we proceed to demon- 

 strate. Writing either k/n or 1 - k/n, as K, we have 



1 If the coefficient E in the term Ez* is essentially positive it gives a 

 slight excess in volume or area : the proper coefficient in the expression 

 of that quantity being ± Ez 5 , while the formula gives \\Ez 5 . HE be 

 small the difference ^-f^ Ez 5 may often be negligible. 



