132 G. H. KNIBBS. 



4. Volumes of warped-surface solids. — The area of any polygon 

 of n sides, expressed in terms of the coordinates of its angular 

 points, is 



il = 42?[(a^+i-ai_ 1 )y k ] (6); 



consequently the volume of a solid whose parallel end-planes, the 

 perpendicular distance I apart, are polygons of the same number 

 of sides, and whose mantle consists of warped surfaces, is : — 

 V=-hl {2?[(^+i"a^)(2y k + y' k )] 



+ ^[(^ +1 -x^_ 1 )(2y^ + y k )]} (7> 



as may easily be seen by forming the area of the middle section. 

 When a series of such solids are contiguous, so that the end planes 

 are common to the adjoining solids and the distances between 

 those planes are equal, this formula, (7), maybe thus extended: — 

 Put for brevity 



^k = (%+i - ^k-i) ; XL = (a'k+i - a'k-i) ; (8) 



and so on : then omitting the suffixes since no confusion can arise, 

 and omitting also the limits as they are obvious, we have 

 V m = -&l{2[X(2y + y')] + ?[X'(y + 4:y' + y'')] + ... 



... + 2[j: L ( 2 / K + 4 2 / L + ^ M )] + 2[X M ( 2 / L +2 2 / M )]} (9) 



in which K l m are to be understood as accents merely : V m is the 

 total volume of the m solids. The initial and final terms are 

 identical in form, and all interior terms are also identical with one 

 another. This last expression reduces the scheme of computation 

 to its simplest form, : its prismoidal character is apparent. 



5. Solids of trapezoidal section with two warped surfaces. — In 

 certain cases which present themselves practically for solution, 

 the formula may be greatly simplified, In Fig. 2 let the heavy 



lines denote the parallel 



end areas of a prismoidal 



figure, the sides AB = a, 



A'B' = a', CD = b, and 



CD' = b' being parallel 



to one another. The 



surfaces AA'C'C and 



B B'D'D will be warped, 

 Fig. 2. r 



