342 EEPORT— 1880. 



A second application of the rules to either set gives 



«! + 4^1 + 

 + 4^2 + 1652 +4c2 

 + ^3 + 4^3 + 



It might be supposed that this represented the volume-integral of a 

 paraboloid with its axis parallel to the ordinates ; but this is not so, for in 

 the paraboloid, since all parallel sections are equal and similar, 



ai — 2&1 + Ci = ttj — 2Z)2 + a2 = a3 — 2&3 + c^ 



or f tti — 2bi + Ci ") 

 ; _2a2 + 46., -2c2 > = 



Combining this with V (above) gives 

 9 f &, + 



S [ + h, + 



So that the volume of the paraboloid for nine ordinates is given by either 

 set of five symmetrical ordinates only : that is to say, by either the central 

 one and those at the four corners, or by the central one and the four at 

 the middle points of the sides of the square. 



Dr. "Woolley, to whom this simplification is due, showed that this rule 

 applied not only to one paraboloid through the heads of the nine ordi- 

 nates, but to the sum of the volumes of two paraboloids in two ways, 

 either 



or 



In fact, let h.^ be taken for origin, a^ h^ 0-2 for the axis of x, and h^ h.^ 63 for 

 the axis of 1/, 2 being normal to the plane of the paper, then writing the 

 equation to a paraboloid as 



z ■= a + hx + cij + dx^ + ey^ + fxtj + ax^ + fix'^y + y xi/ + h/ 



and integrating first with regard to y between the limits +— x, and 



then with regard to x between the limits and h, the volume whose base 

 is the triangle Cj &2 ^'s is expressed by 



alih + % hhV: + I d/M+l eJiP + ^ ah*]c + -yh^lcK 

 o Z K) o 16 



The other three components may be obtained by interchanging li and Z;, 



and other corresponding letters, and then by changing the signs of/i andZ-. 



Tlie altitudes of the nine points are obtained by writing and + h for 



* See the JLoJianic's 3Iagazinc for 5tli April, 1851, vol. 54, p. 265 ; also Murrmfs 

 SMphinldino, pp. 35-6. See also Inst. JS'ai: Arch. vol. vi. (1865), p. 4i, and vol. viiL 

 ("1867) p. 210. 



