66 G. H. KNIBBS. 



The following table shews the coefficients obtained by combining 

 in different ways equations (61) to (65). 



XIII. — Integral expressions for volume, Jive symmetrical sections. 



Indices p cr aore /? or S y 



1, 2, 3, 4 and 5 90 7 32 12 



1, 2, 3 „ 6 8190 629 2944 1044 



1, 2, 3 „ 7 1050 79 384 124 



1, 4 „ 6 69510 5323 25088 8688 



1, 4 „ 7 9730 729 3584 1104 



1, 5 „ 6 44730 3389 16384 5184 



1, 5 „ • 7 8190 607 3072 832 



1, 6 „ 7 9310 679 3584 784 



Still further, if n = 5, the general equation (57) becomes 

 /5|(^ + l)(l + 4iV 2 - 5l j+7((^ + 1 )( 2P+3P )- 2 - 5P } + (p- 1 ) 5P = 0...(67) 

 giving the following solutions for y : — 



Index;; = 2 or 3 y = f T + - r \- /3 (68) 



p = 4 y = itfL + ri T p (69) 



^ = 5 y = W - A-jS (70) 



p = 6 y = ffifl - AWr /? .... (71) 



From (68) and (69), (68) and (70), and from (69) and (70) we find 

 ifi — -ff-, 7 = yf, hence these are conjugate indices: (68) and (71) 

 however give /^= ffff. Hence we see that (67; with the indicated 

 weights satisfies only a quintic function, and not a sextic. Hence 

 if the transverse section of a solid is known to be a quintic function 

 of the distance along its axis, five sections, that is two terminal, 

 a middle, and two other equidistant sections are sufficient, and 

 there is no advantage in taking six sections. Thus the function 

 beinfj quintic, as before following (66), the volume is exactly given 

 by the formula in Table IX., V — yv s - z (iL.fiM), //, denoting any 

 coefficient and M the corresponding section. 



As before, an infinite number of formula can be developed for 

 p— 1, 2 and 3 ; for example, any of the following series of formulae 

 satisfy a cubic function. 



