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1144

Fig. 20 shows the resulting nomogram constructed with

Ml = jM2 = -2M3 = M4

Suppose now that we want to know the rate of discharge from a loo-yd. pipe of l-in. bore, with a head of water of I in.

(C)

1000 ft/hr

600 400 300

25 2(5

< -10 "

06

Fi r. 20.

a i

1000-" 9 8

7 6

500'-] 4 3

50".

Join I on the (D) scale to I on the (H) scale. Join the point where this straight line cuts the reference line to 100 on the (L) scale, and produce the straight line backwards to cut the (Q) scale at 201 ft./hour (see dotted lines, fig. 20), the required rate of discharge.

These Double Alignment Nomograms can be constructed by com- bining any two of the types, A, B, or C where the four-variable formula can be written in the appropriate form.

Take for instance the formula

H= 18,400 (log Bi - log B 2 ) (l + 0-003676)

giving the difference in level (H metres) between two stations at which the barometric readings are Bi and B a mm. respectively,

the mean temperature being 0C. ( = - J Writing it

TT

18,400 (I + 0-003679) l l

it can be broken up into two partial nomograms of Type B and A respectively.

H*l8400(logB,-logB,-{l*0-003670) 77

760

(B,)

770 mm

760

750

700-|-500 (HI

9

-1000

-L600 Fig: 21.

Fig. 21 shows the resulting nomogram, and, to illustrate its use, suppose 61 = 750 mm., 62 = 670 mm. and it is required to find H.

Join 670 on (62) to 750 on (Bi), and produce to cut the reference line. Join the point thus obtained on the reference line to +20 on (8 ). This straight line produced will cut (H) at 935 mm. (see dotted line, fig. 21), the required difference in height.

(ii). Combination of an Alignment Nomogram with a Network.

Suppose we have a network (zi, Z2), com-
 * posed of two systems of figured curves (zi),

fe) crossing each other (fig. 22).

If we take any point on this network, a curve of both systems will pass through this point, and we may assign to the point a value of both Zi and of Zj, taking the values from the curves of the systems (z_i), (zj) which in-

' A \\\ \ tersect in the point. The point has thus in a

sense two values and is termed a binary point. The general equation in parallel coordi- nates of such a binary point will be of the form

and its coordinates in cartesians with the usual axes will be

X := \ "V ^ "

hl2\ KiZ "12 f l2

We can obtain the equations of the systems (zi), fe) forming the network (zi, z 2 ) by eliminating in turn Zi and Zi between the above expressions for x and y.

Consider now a formula that can be put in the form

This can be represented by the rectilinear parallel scales

and the network

As an example take the Compound Interest Formula

M = PR" where P is the principal, M the amount, Rthe amount of l for i

year at r % per annum ( i.e. R = I H -- J, n the number of years.

Writing it

log P + n log R log M = o and taking

2i = P, /i=logP z 2 = w, /2 = n Zs = r, /34=-logM z = M, 34=1, A 3 4 = logR

the nomogram will consist of the parallel scales M = Mi log P

log M

MlM2 log M

w log

and the network (r, M) defined by ttiu+iii log RD or in cartesians

_^Ml log R M2

Mi log R+M2 1

The expression for x is independent of M, so that we have (r) a system of straight lines parallel to (P) and (n).

For the system (M) we have, eliminating R between the abov expressions for * and y,

2\y=m log M (8*)

hence (M) consists of straight lines radiating from the point * = X, y=o (i.e. the zero of the n scale), and cutting the straight line 3t= X (i.e. the P scale) at the points

y= MI log M

so that the lines of the system (M) are easily drawn from the gradu- ations of (P).

9

20 -|

8

7

V

6 5

\

.

^

\

\

,

x

>

",:"'*.

10

g

4

\^

1

g

/PI

s

..

s



7 9 .... -10-

3

V

r

s

6

--

IMI

S

.

4

2



3

++,

5-



2

1

1 0.

Fi r. 23.

Fig. 23 shows the completed diagram. Suppose, for instance, we want to know the amount of 300 in 10 years at 5% compound interest. Joining 3 on the (P) scale to 10 on the (n) scale, this straight line cuts the 5 % line (see dotted line, fig. 23) at a point corresponding to the line 490 of the system (M).

BIBLIOGRAPHY. M. D'Ocagne, Traite de Nomographie (1899); Calcul Graphique el Nomographie (1908); Principes usuels de Nomo- graphie avec Application a divers Problhmes concernant L'Artillerie et L' Aviation (1920); Lt.-Col. R. K. Hezlet, Nomography (1913); J. Lipka, Graphical and Mechanical Computation (1918); C. Runge, Graphical Methods (1912). (R. K. H.)