Mr J. Sang on a Systematic Method of 



prismoids subtracted from the sum of the other set. But the 

 solidity of any of these prismoids is equivalent to its base, 

 multiplied by the third part of the sum of its parallel sides. 

 The base itself is equivalent to half the latitude of each point, 

 multiplied by the longitude of the point before, minus the lon- 

 gitude of the point after it. Again, this base is a negative or 

 positive quantity, according to the direction in which we pro- 

 ceed around its boundary ; and if we conceive ourselves to be 

 placed on the outside of each face of the solid, and to proceed 

 always in one direction, say from left to right, the signs of the 

 areas of the faces in the upper surface will be contrary to the 

 signs of the areas in the under surface. Therefore, if the solid 

 is regarded as composed of two sets of prismoids, whose bases 

 are determined by the law of co-ordinates, and taken always in 

 the same direction, its capacity will be simply equivalent to the 

 algebraic sum of these prismoids. 



Assuming a specific solid, as the hexahedron A B C D E, and 

 indicating the latitudes of each point by the large letters A, B, C, 

 &c. ; the longitudes by the small letters a, 6, c, &c. ; the alti- 

 tudes by the letters a, /3, y, &c. ; and the solidity by S, we will 

 have, 



C(a-6) 



C(d-e) 



cs = 



/ (A(c e)) v , (D(5 0) 



\JC (e a) U + y + )+ ( 1 B (e d)l(S + + ) ) 

 (E(ac)j ' > (B(rf_i)j 



wJAoL! 



V (E(6-a) 



which equation expanded and simplified becomes, 



'B(c - 



c s = 



a) 



A(c b) ) 



(* c)\ , /B(a d)\f 

 [b-J>)+'(c(d-aV\ 

 (c~6) D(6_e) J 



And it is evident that the formula may be applied to any rec- 

 tilinear body in these words, six times the solidity of any recti- 



