54 G. H. KNIBBS. 



Prop. (k). If the coefficient 4 be assigned to the middle section, 

 the indices satisfied will be 1, 2 and 3, and none other. 



The formula for volume or area in the case above considered^ 

 and when the original function 



A Z = A + Bz + Cz u + Dz\ 

 u and v as before being conjugate ; is : — 



V^^z^ + pBn + C) (38) 



the subscript denoting terminal sections, and m a middle section. 



13. A middle section, and two other sections equidistant there- 

 from, all of equal weight. — In this case, if each section have unit 

 weight, (5) reduces to 



a p + (l-a) p = _A- -JL (39) 



v ' p + 1 2 P v ' 



which is clearly analogous to (236) § 9, and may similarly to (22) 



be written 



p(p-l) p( P -l)( P -2)( P -Z) e < + etc ,.. = 3/_L_ 1\ , 40) 



S A ! OP-4 S ° oL,.l OP V I 



2! 2 p - 2 * 4 ! 2 p ~ 4 * 2\p+l 2 P / 



By these equations the results in Table VI. are calculated; the 

 curve is shewn in Fig. 2, Curve 3. 



VI. — Position of two sections equidistant from a middle-section. 



Index. a Index. a Index. a 



-1121500 1 3 -1464466 3 10 -1221566 

 •25 -1221114 4 -1439328 11 -1185688 

 •5 -1295311 5 -1406342 12 -1151021 



1 -1391506 2 6 -1370589 13 -1117833 

 1-5 -1442089 7 -1333554 14 -1086176 



2 -1464466 3 8 -1295980 4 15 -1056064 

 2-449 -1469624 9 -1258477 oo -0000000 



Note. — b — £ ; c = 1 -a. The coefficient of each section is unity. 



1 The limiting value may be found as previously shewn : the equation 

 1S log e a-— -—-.--—- etc. = -3 + log e 2= -2-3068528. 



2 The equation for the limiting value is : — 



a loffe «-«+ — + a l + ... + — — + etc. = £ log e 2 - f = - -4034264 



1.2 2.3 {n-l).n 



See § 16 hereinafter for a fuller consideration of these limits. 



3 Both roots, that is forp=2 andp=3 are the same, viz., a = %-\ */2. 



4 For seven places of figures, we may, after_p = 7, put a p =0, conse- 

 quently 6p = 3/ (p + 1) - 1/2P . 



