APPLICATION AND DEVELOPMENT OF PKISMOIDAL FORMULA. 137 



V m = -^r I | iv (3c + 6 <?!+ ... + 6 c k + ... +6 c m -i + 3 c m ) + 

 « o (2rf o + rf0+-.-+«k(rfk-i + 4et + <4+i)+..-+«m(^m-i + 2rf m ) + 

 <(2o? + dj) + . . . + e k «-i + K + o? k+1 ) + . . . + e m (d m -i + 2C)}(20) 



in which the values of & are 1 tom-1. The scheme of numerical 

 computation is obvious : the process cannot be further abbreviated. 1 



1 0. Approximate estimate of volume from profile of centre-line of 

 m longitudinally contiguous solids. — If the centre-heights of the 

 end planes, of the form ABGF Fig. 3, be given, the values of r 

 and w being constant, the error of treating two adjoining solids 

 as a prismoidal figure of the length 2 1 will in general be small, so 

 that approximately, m being an even number, the total volume is 



^=i^^ + 4 gi Cf+2 ?2 ^ + 4^^ + ...+g m ^ n -tm^)...(21) 



in which q is unity if the surface is level across, see § 7, and C is 

 formed as indicated in (12). 



11. Solids whose longitudinal axes are curved. — -Let OY be the 

 line of intersection common to three planes OX, OP, OQ ; to one 

 of which — OX say — the axis of a prism, whose right section 

 thereon is any closed curve whatsoever, is vertical ; the prism 

 being cut also by the other two planes, making the angles 6 ± and 2 

 with the plane YOX. Then the intercept between the PQ planes, 

 of any line parallel to the axis of the prism, is 



I = z 2 - £j = x (tan 2 - tan ± ) — \ix say (22): 



OX and OY being the axes of the abscissae and ordinates, to the 

 plane of which z is perpendicular, hence the volume of the portion 

 of the prism intercepted between the P and Q planes is 



V=ff f xxdxdy (23), 



and since the area of the right section of the prism is 

 A=ffdxdy (24) 



1 See the suggestions in the footnote to the previous section, § 8, relat- 

 ing to diagonal additions, etc. There is no advantage to be derived from 

 continuing the lines GA, HB to J as in the previous case. The express- 

 ion for the total area merely requires that c in (19) be replaced by C - DJ; 

 see § 14 hereinafter. 



