VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 



51 



The integral solutions up to the third power inclusive are exhibited 

 in Table IV. hereunder. 1 



IV. — Positions of first section; the second being the terminal 

 section. 



Indices. a = Distance along j3 = coefficient 



p q axis. of B 



% 1 0-2500000 0-5000000 



1 2 03353333 0-3333333 



1 3 0-3660254 0-2679492 



2 3 0-4215352 0-1692577 

 2 4 0-4472136 02000000 



General coefficient 

 1/U+0) 

 •6666667 



0-7500000 

 0-7886751 

 0-8552434 

 0-8333333 



The formula for volume or area will of course be 



(34) 



l+£ 



(A, + pB ) 



A x denoting the area or ordinate at a, and B that at the terminal. 

 For jo = 1, q = '2, we may instead of (30) and (34) write 



7=l*(A + 3Bi) = lz(3Ai + B ) (25) 2 



A denoting as before tho initial section or ordinate : this reciprocal 

 symmetry does not extend to other cases. Some of the formulae 

 of III. and IV. may be expressed in the following forms : — 



/T=J; 0=1 :— V=iz(A +8B_ g _) 



V=±z(2A i + B°) 

 p = 2; q=4=:— V=$ % (4^+ 5B^) 



V=iz(5A L + B ) 



V 5 



(35a) 3 



in which the suffix indicates the distance along the axis from the 

 A end. By means of (28), (29), (32) and (33) it is easy to develope 

 similar expressions to these last ; they would however probably be 

 of no practical moment, and are not here further considered. 



1 The equations equal to zero, and the roots are as follows : — 



. q a Equation. £ Equation. Values of a and £ 



.2 a a -|-a + i=0; p = I -2a = i (I -3a 2 ); a = i; (3 = i 



1 .3 a 3 -f a + i=0; £= 1 - 2a = i (1 - 4a 3 ); a = £(V3-l); = 2- V3 

 2.3 a 3 -|-a 2 + |=0; 0=i 1 - 3a 2 ) = i(l -4a 3 ); a =-± 6 -(l+ V33); fi = 



-As (77-3^33) 



2 Kinkelin's formula. — Grunert's Archiv., Bd. xxxix., pp. 181-186, 1862. 



3 Puller's formulae. See ' Erweiterung der Prismatoidformel/ — Zeit. 

 fur Vermess., Bd. xxix., p. 36, January 1900. 



