438 
ME. E. J. EEED ON THE UNEQUAL DISTEIBUTION OF 
of the weights such as keep the number of balanced sections constant, of course applies 
with equal force to the sections of maximum and minimum bending-moment. In both 
tig. 11 and fig. 12 (Plate XVII.), for example, we have six balanced sections (IP IP, IP IP, 
&c.), but only three sections of maximum and minimum bending-moment in the latter 
instead of five as before ; while we have seen that another slight transposition of weight 
would leave only one section of water-borne division and maximum bending-moment. 
A comparison of the curves of bending and shearing in these two cases possesses further 
interest, on account of the fact that when the curve V V in fig. 12 has a minimum posi- 
tive ordinate at IP IP instead of a maximum negative ordinate as in fig. 11, the maximum 
and minimum ordinates of the curve M M at e d and d d' respectively in fig. 11 disappear, 
and we have instead of them a point of contrary flexure in the curve M M of fig. 12. 
Similarly, if the other transposition were made which does away with the two sections of 
water-borne division a a! and b b' in fig. 12, and turns the small positive maximum shear- 
iug-force at IP IP into a small negative minimum shearing-force, we should have a point of 
contrary flexure at IP IP instead of the maximum and minimum ordinates at a a' and b b' 
of the curve M M. In fact, from the relations which exist between the curves V V and 
M M, it is obvious that at the balanced sections, where the shearing-force has maximum 
or minimum values, the curve of moments has either points of contrary flexure (as in 
all the illustrations we have given) or singular points where there is a change of cur- 
vature. It may be added that we have by our construction supposed the points of con- 
trary flexure in the curve of loads to lie at the balanced sections where the curve crosses 
the axis ; and hence we may say that at the stations where the curves L L have maximum 
ordinates the curves V V have points of contrary flexure ; in other words, the curve V V 
bears a relation to the curve L L similar to that which the curve M M bears to it*. The 
broad practical deduction to be drawn from the cases represented by figs. 11 & 12 is, 
however, simply this, that by transposing weights from the centre to the extremities we 
render the curve of moments M M less tortuous, which is a matter of no consequence ; 
but we at the same time increase the maximum bending-moment which the ship has to 
resist, and which may thus become raised to a very undesirable amount. We are thus 
again reminded of the reduction of the strains of ironclads produced by adopting the 
belt-and-battery system instead of that of complete protection. 
In all ships there must obviously be a section or sections of absolute maximum bending- 
* Expressed in mathematical symbols, this relation stands as follows : — Suppose B to be the origin of coor- 
dinates in fig. 12, and let the distance of any station from B be called x. Then if ?/= resultant vertical force, 
or ordinate of the curve of loads at that station, S= shearing-force, or ordinate of the curve of shearing, and 
M= bending-moment, or ordinate of the curve of moments, we have by our method of construction, as pre- 
viously explained, 
s =Jc *y- dx > 
M— . xdx= j* S . dx. 
That is to say, we obtain the ordinates of the curve of shearing by integrating the areas of the curve of loads 
up to various stations ; and obtain the ordinates of the curve of moments, either by integrating for the moments 
of the curve of loads or for the areas of the curve of shearing. 
