ON QUADRATURES AND INTERPOLATION. 345 



it is to be observed that, -wLile the first is less accurate on tbe supposition 

 that the surface is of strictly parabolic character, and convergent, yet it has 

 the advantage of taking account of the surface (using the above example) 

 at 45 points instead of 30. It thvis secures that the surface to ■which 

 the arithmetical summation refers shall coincide with the surface to be 

 measured in 45 points as against 30, on the assumption of accurate 

 measurements. The advantage of the higher rule, therefore, depends upon 

 there being no possibility of a pei'iodic term, and upon thei'e being no such 

 "want of convergence as would render terms of higher degree than x^ y^ 

 noticeable. If the ordinates are inexact, this advantage of the polygonal 

 rule holds a fortiori. 



The author has shown* that there exists a similar reduction in the 

 number of ordinates necessary for the summation of a triple integral. 

 Writing the 27 ordinates of 



u = Oq + a^x + ft^y + y^z 



+ 020;^ + ftill^ + 72*" + ^y^ + 1-'^^ + ''•'«Z/ 

 4- a^x^ + jS^y^ + 732=5 



+ (^l2 + ^22/) y^ + (a'i^ + j"2-) ^^ 



+ ("2-^+ ^'iy)xy 

 as 



oi hi ci a/ 6/ c/ a^" h^" c/' 



a2 &2 ^2 ^2' ^2' '^2 C-2" ^2" ^2" 



«3 ^3 C3 «3' ^3' C3' a^" I3" C3" 



the treble integral / , I -i- I 1 ^^ '■^'^' ^^^J ^'~ '^^ expressed by 

 -| hlclih^ + li' + a,' + c,' + h,' + h,") 



in which the absolute middle ordinate does not appear. In fact, arrang- 

 ing the 27 letters which represent the ordinates in a ctibe, the only ones 

 which appear are the middle ordinates of faces. The late Professor 

 Rankine expressed this rule in the following form : ' The mean density 

 of a parallelepiped is the mean of the densities at the middle points of 

 its six faces.' This supposes the density to be a parabolic function of the 

 three co-ordinates, not higher than the third degree, and thus, of course, 

 excludes the case (which usually presents itself physically) of the density 

 varying from the middle to the bounding surface. This rule, like 

 Woolley's, may be modified by using- corner ordinates, or the ordinates 

 corresponding to the middle points of the edges : only then the formulae 

 are less simple, and the middle ordinate of all does not disappear. 



All the remarks about Woolley's rule inadequately representing the 

 surface, as compared with the polygonal rule, apply a fortiori to this. 

 Whatever may be the convergence, except upon a certain limited hypo- 

 thesis, namely, that the function is of definite parabolic form, coincidence 

 between the actual subject of integration, and the subject of summation, 

 is secured at too few points for the results to be i-eliable. 



It had been observed by the author that there was a peculiar relation 

 of the ordinates in these rules, namely, 



* See Scott Eussell's Modern A'aval ArcJdtecture, vol. i. p. 127, and Trans. I.N.A. 

 vol. vi. (1865) p. 47. 



