V=ji(A+£z + Cz*fl + 



144 G. H. KNIBBS. 



integration the expression for the volume will be higher than a 

 quartic function of z, or will contain logarithmic functions. For 

 example,, if in (33) R be a function of z, 1 that is if the centre-line 

 be not a circular curve, let 



R z = R (l + a z + (3z'> -fete.) (38); 



then remembering that R z d6 = dz, the expression for the volume 

 will become, including also a non-linear variation of the abscissa 

 of the centre of gravity of the right-section 



x + Xz + I** + etc. | 



R (l + az + (3z* + etc.) J I { ™ } 



For the purposes of practical computation there is no special 

 difficulty in integrating this for particular cases : in general it 

 may be said that it will be convenient to expand the expression 

 in the square brackets in the form of a convergent series, and thus 

 to integrate only the terms of sensible magnitude. Thus the 

 above expression can, in practical cases, always be put in the form 



V = j[ (A + Bz + Gz*) 1 + -L (x + az + bz* + etc.)] icfe...(40) 



in which with the above coefficients 



a = (X- ax) ; b= [/x- a A + (a 2 - /3) x] etc (41). 



The value of the integral is obvious. 



17. Suggestions respecting the use of the preceding formula. — 

 By the use of suitable tables, the calculation of which is very simple, 

 and the nature of which has already been sufficiently indicated — 

 see §§ 7, 8, 15 and footnotes — the applications of the preceding 

 formulae can generally be made to involve nothing more than simple 

 additions and subtractions ; as will at once be evident to any 

 computer : it is unnecessary to enter into details. 



In respect of corrections for the positions of the centre of gravity, 

 formula (36) it may be observed that if x and x v are approxim- 

 ately equal, the factor 



i x n + x, 



1 + -T7T 



1 As it is for example in so-called 'curves of adjustment' in railway 

 location. 



