166 The N.Z. Journal of Science and Technology. [Aug. 
particular site. They will, of course, vary enormously for different sites. 
The volume ratio K -y k is a useful criterion of the value of a site, a high 
value being favourable, and vice versa. If a trial survey gives a figure of 
500 or over, full investigation would usually be warranted. 
When a shallow stream is improved for navigation by weirs and locks, 
without any consideration of the use of the structures for other purposes, 
it may be shown that the most economical development is obtained by a 
succession of quite low weirs ; for the cost of a weir is in proportion to 
some power of its height, while the length of the reach formed by the weir 
is proportional to its height merely. Thus suppose a length of M miles 
falling uniformly/feet per mile to be deepened by a series of n weirs (fig. 1). 
Let a = natural depth, and 6 =. required depth of water, c = height of 
backwater curve due to sensible velocity in upper part of reach, and x = fall 
in the bed in one reach from weir to weir. Then H, the height by which 
M f 
the v T ater-levei is raised at the weir, = x + b — e — a, and n — —. In 
x 
broad shallow streams with well-defined banks the length of a low weir is 
not much altered by variations of its height, and so the cost is proportional 
to the square rather than to the cube of the height, up to a certain point. 
Approximately, therefore, the cost of each weir = MI 2 , and the cost of the 
M f 
whole improvement = nJcH 2 = Jc — (x + b —c — a) 2 . By differentiating 
and equating to zero the cost is a minimum when x — b — c — a, giving 
H = 2x and height of lift at weir = H + a — b. For river-barge work 
a suitable value of b is 5 ft.; if in a particular instance a is 2 ft. and / is 4 ft. 
per mile, c will be about 0*2 ft. ; then x — 2-8, H = 5-6, and lift of lock 
2-6 ft. By trial of other values of x, however, it is seen that there is not 
much variation in cost until x is more than double the value obtained by 
differentiating ; and, as practical considerations are in favour of the higher 
figure, let x be taken as 5*6, when H = 8-4 and lift of lock 5*4 ft. In 
England higher weirs than 6 ft. or 8 ft. can seldom be adopted without 
embanking the rivers to prevent flooding of adjoining plains, but in New Zea¬ 
land, where many of our rivers are deeply entrenched, it will often be 
advantageous to use higher weirs and gain the benefit of longer reaches and 
less frequent locks. 
When the summer flow is so small as to be negligible, a and a are zero, 
and the cost is a minimum when x = b : that is, the lift at each lock is 
equal to the required depth of water. Here again there is very little 
increase of cost until x = 26. 
When the question of utilizing the weirs for power-production is intro¬ 
duced, on the basis of the summer flow without storage, their initial cost 
is again least when they are low and numerous, seeing that the power given 
by each is proportional to its height, while the cost varies as the square or 
cube of the height. But the economics of power plant and operation 
strongly favour reducing the number and increasing the height of dams, 
and in any case a height would be adopted which would permit the turbines 
and generators to be direct-connected without resort to extremes of design 
—say, not less than 30 ft. 
If now the utilization of the storage capacity provided by the dam be 
taken into consideration, the economic height to which a darn on any given 
site may be built is likely to be largely increased, seeing that capacity and 
cost are (roughly speaking) simply proportional. Also, the relative capacity 
to be obtained at different sites varies so considerably that it will generally 
