344 



EEPORT — 1880. 



its base, and AB for its altitude. Hence the volnme of the paraboloid 



between the extremities of nine ordinates, a^ C3, resolves itself 



into 2Z;.i^, the parabolic area whose ordinates are a., b^ c^, added to 2/i x the 

 parabolic area whose ordinates are 



i, - h.2, b. - bo(=0),b3 - b.2 



or, by the rule for parabolic areas, 



V = 27c I («2 + 4^b, + c,) + 2h I I (b, -I,) +4. (b.^ - b.,) + (b, -bA 



2 

 = - Mc (a.2 + by + 2?)2 + is + Cg) 



The restriction as to the direction of the principal planes is equivalent 

 to writing F = in the general equation of the paraboloid 



Z=A + Bx + Cy + D.r2 + Eif- + Fxij 

 but 2 (x + p) (x -t- gr) = 4 xij identically, and (x + p) (x + q) — xij is 

 the variation of any ordinate (p, q) from the middle one, as regards this 

 term alone ; it is therefore evident that, for the symmetrical integral, the 

 effect of this term vanishes. 



Two applications of the polygonal rule are easily seen to be equivalent 

 to drawing a hypei-bolic paraboloid through the heads of every four ordi- 

 nates, the four right lines joining the ordinates two and two being the 

 generators (of both systems). The last paragraph shows that its volume- 

 integral can be expressed by interpolating a middle ordinate, and using 

 that only, instead of the four others. This appears to be equivalent to 

 the reduction obtained by Woolley's rule in the degree above ; but it is 

 of no practical use, seeing that it only substitutes the sum of (r?i — 1) (?i — 1) 

 ordinates for that of mn ordinates, an advantage which, in general, is no 

 compensation for the interpolation. 



Two applications of the polygonal rule lead to the scheme of multi- 

 pliers : — 



111111111"^ 



2 

 1 



2 2 2 2 2 

 1 



111 



2 

 1 1 



4 



1 



2 



1 



2 



1 



2 



11111111 



42222222 



AA; .^ ^ 1 1 1 11 11 i > 



1111111- 



on comparing this with Woolley's multipliers 



± 0- o-" Oi 



hk< 



2 2 

 1111 



2 

 1 1 



10 10 10 



1 



± 



1 1 

 1 



2 2 



1111 



0- 0^0 

 2 2 



