VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 59 



A 2/ = 2/n-™/n-i+ 2 ! 3/n- 2 - etc (47) 



so that we have only to insert the proper line of coefficients from 

 Pascal's triangle and reduce. In this way the following formulae 

 are obtained. 



IX. — Weight-coefficients with terminal and equidistant inter- 

 mediate sections. 



V=iz(A + B )=±z(A + iB m+ C ) = iz(A + 3B + C + D ) 

 = ^o-z(7 + 32B+V2C + S2D + 7U ) 

 = 2-b z(\9A + 75B + 50C + 50D + 75E + 19 F ) 

 = ^o(^A + 2l6B + 27C+27'2D + 27E+2\QF + UG o y 



The deduction of formulae for volumes or areas by this method 

 does not fully reveal the sphere of their legitimate application. 

 For example the first formula in IX. is legitimate only when the 

 sectional-area linearly changes with the distance along the axis : 

 the second formula is deduced on the necessary assumption that 

 the sectional area is a quadratic function of the axial distance ; 

 it proves to be absolutely correct also when that area is a cubic 

 function. The third formula is derived by assuming that the 

 sectional area function is cubic : it is a good approximation, even 

 when the function is quartic, but is not exact, since it involves an 



1 " Weddle's " rule is merely an approximation. The exact expression 

 in difference-terms is 



V=z(y + ZA { y + 41 A n y + 4A m i/ + ±±A iv y + ^A y j/ + ■#& & vi v) 

 If, in this, the coefficient -gVo be changed into -££$, and if moreover the 

 proper coefficient of D is diminished by -^\, that is if |f§ be put for §-£§. 

 in the above formuU, then the formula may be simplified into Weddle's 

 approximation, and written 



V= -i^z (A + 5B + C + 5D + E + 5F + G ) 

 The statement in the Encyclopaedia Britannica 9° Edit, xvi., 22, would 

 be less liable to mislead if it read " approximate formula for the area " 

 instead of " formula for the approximate area." The statement that the 

 formula is derived in the manner indicated is moreover inaccurate. It is 

 obtained by a purely arbitrary proceeding. Prof. Johnson's statement, 

 in his " Theory and Practice of Surveying/' p. 610 Edit. 1887, that, if the 

 coefficient g^j be changed in the manner indicated, Weddle's rule may be 

 obtained, is also not accurate. The expression is not exact even when 

 the sixth difference is zero. 



