58 



G. H. KNIBBS. 



VIII. — Two terminal sections and one intermediate, all of equal 

 weight. 



Index. 



6 



Index. 



b 



Index. 



b 



Index. 



b 



•5 



1 -00000 



9 



•54484 



1-3 



•40049 



1-7 



•27459 



•6 



•80047 



10 



•50000 



1-4 



•37150 



1-8 



•23081 



•7 



•68165 



1-1 



•46289 



1-5 



•34200 



1-9 



•16995 



•8 



•60240 



1-2 



•43042 



1-6 



•31041 



2-0 



•ooooa 



The curve is shewn in Fig. 1, see Curve 5. 



The general results of this section may be summed up as follows : 

 Prop. (n). When the weights of three sections, viz., two terminal 

 and one intermediate in a definite position, are deduced so as to 

 satisfy two indices, a third conjugate to these will in general be 

 satisfied: two of the indices, or all three, may, with particular 

 values for the weight-coefficients, become identical. 



15. General result of the method of finite differences. — If the 

 axis z be divided into n equal parts, and sections be taken at the 

 terminals and points of section, the areas of these may be repre- 

 sented by the n + 1 ordinates thereat. It is always possible to 

 draw through the n + 1 ordinate-terminals a curve of the nth 

 degree, so that if the order of the surface is really represented by 

 that curve, the indices in the original function viz., p, q, r, etc.,. 

 are merely 1, 2, 3,... n. 



By the calculus of finite differences it is shewn that the volume 



or the area, may be readily expressed in terms of the first rank of 



differences, and n, the number of parts into which the axis is- 



divided. To a curve or surface of the sixth degree the equation 



is, y denoting the first ordinate, and A { y, A i{ y, etc., the first rank 



of differences, 



'^+ln 2 A i 2/+-i-(2n 3 -3n 2 )A ii 2/+TA-(^ 4 -4n 3 + 4n 2 )A iil 2 /l 

 io(Qn 5 - 45n 4 - llOn 3 -90n 2 )A iv y [^ 



V= 



45n 4 -110n 3 - 

 + T -A-o(2rc 6 -24rc 5 + 105™' 



200n 3 + 144n 2 )A v 2/ 

 4725rc 4 + 7672rc : 



o (12n 7 -210rc 6 + 1428n J 

 -5040rc 2 )A vi 2/+ etc. 



This maybe recast in terms of the ordinates themselves by observing 



that the coefficients thereof, connecting them with the differences,. 



follow the law of binomial development ; that is, 



