ON QUADBATURES AND INTERPOLATION. 



343 



X and 0, and + Ic for y. Making these substitutions, the volume of tlie 

 whole solid is found to be 



V = I Ait (6a + Uli^ + 2e^2) 

 o 



= |M(a2 + &i + 2&2 + J3+C2) 



= Q ^^' («1 + «3 + 862 + C3 + Cj). 



If, moreover, the paraboloid be reduced to one of the second degree by- 

 making a/3yo vanish, the following equations also hold: — 



2 2 



vol. on tti ci C3 = I Ilk (6i + &2 + C2) = 5 /iA; (2^2 + "i) 



|/A^^^ 



o 9 2 



vol. on ci a, 03 = ^ 7iA; (^2 + ^2 + ^1) = 5 ^^-^ (2&2 + »i) + g/A^A;^ 



vol. on ai as C3 = I TiZ; (^2 + &2 + ^3) = | ^^ (2^2 + ^a) - 1 /^^'^' 



vol. on ci C3 a3 = I AZ; (63 + ^'2 + ^2) = | ^^ (2&2 + C3) + g/^"^' 



The rule for the corner ordinates is not very convenient. The other 

 rule, when we have nine ordinates only, may be written, having regard to 

 the coefficients alone, as 



010 

 121 

 010 

 For a considerable number, say 5 X 11, it becomes 



01010101010 

 12222222221 

 02020202020 

 12222222221 

 01010101010 



The rule for the coefficients is that all the ordinates which are odd in both 

 planes of section have the coefficient zero : all the others have the co- 

 efficient 2, except the border rows and columns, where the coefficient is 

 1 instead of 2. The summation, governed by these coefficients, has then 



2 

 to be multiplied by - hh, 

 o 



A geometrical proof is easily given, as follows. Let 

 abed be a portion of the paraboloid corresponding to the 

 rectangular base ABCD, and let the planes of section 

 be supposed (in the first instance) parallel to principal 

 planes. It is a well-known property of the paraboloid 

 that its sections by any series of planes parallel to one 

 another and to the axis, are similar and equal parabolas. 

 Project the arc cA orthogonally on the plane ABia by a 

 cylinder passing through cdcy. It is plain that the solid ^ - j> 

 ABCDcZcyc is this parabolic cylinder i^lus a solid rec- 

 tangle. All the sections of the outlying solid ahcddy, parallel to BCch, are 

 equal and similar portions of equal parabolas, and therefore its volume 

 is the same as that of a cylinder, having the parabolic segment ciy for 







c 



