The expression for * being constant we see that the points are arranged along a straight line Cw parallel to AM, BP, and this straight line cuts AB at a point C such that
CA = _w
CB w
Then writing for the scale (z 3 ) on Cw w= Ms/3 as w is the same as y we have
or
In the particular case where MI = we will have
f.)
_ Ml _ Mi
To recapitulate, the practical procedure is shortly as follows: Select suitable axes AM, Bv and suitable units MI, M2- Draw Cw
dividing AB in the ratio -, and determine the unit n t by the relation
1 = 1+1
Ms Ml M2
Construct along AM, Bt>, Cw, respectively the scales
"=Ml/l
f =M2/2 =M3/3
Any straight line drawn across these three will then cut them at cor- responding values of z\, Zj, z 3 as de- fined by (2).
As a rule we arrange the diagrams so that the scale of the variable, which has generally to be determined in terms of the other two, lies between their scales, as this conduces to greater accuracy in reading.
It is not necessary for the origins A, B, C, to appear on the diagram unless they are required for the range of the variables for which the formula is to be employed. The scales can be quite easily constructed without them, starting from the lowest value required. It will be seen that the freedom of choice of axes and units renders this method an exceedingly flexible one. Exam- ining the range of the variables required for the practical use of the particular formula, we can arrange the scales and the size of their graduations to the best advantage.
Among other things, we wish to avoid the reading straight line making too acute an angle with the scales, as this leads to inaccuracy. Practice will soon enable the best disposition to be seen, but it will most frequently be found convenient to make the useful parts of the scales (zO, (z 2 ) about the same length and about the same dis- tance apart, so that the complete diagram is roughly contained in a square.
As an example of Type A, the formula
fif. 13.
(*)'
already referred to in 2 can be taken, supposing that it is now desired to construct a nomogram to show different values of K, instead of a single curve for a constant value of K. The formula can be reduced to Type A by writing it
logd + |logK-llogC=o iind taking
zf*=<2, fi=\ogd
- -K, /2 = !I ? gK
zs-C, / 8 =-flogC
the scales are all logarithmic scales, differing only as regards their unit.
Take any convenient logarithmic scale that may be available (say that of a slide rule) and by means of it graduate the scales (d) and (K) on two convenient parallel axes (fig. 14).
We can then determine the point C = 10 on the scale (C) by the cross alignments
d = i, K = io
= 2-5, K=8o
for both of which C = 10.
The support of (C) is then a straight line parallel to the axes through this point, and we can graduate it by noticing that for C = K, we always have d = i.
The alignment of d = i with K = 20, 30, 40, 50, in turn, then gives the points C=2O, 30, 40, 50,. ...
Suppose now that we want to know the current which will fuse an aluminium (K=59) wire 0-3 mm. diameter. The straight line
joining 0-3 on the (d) scale to 59 on the (K) scale (see dotted line, fig. 14) cuts the (C) scale at about 9-5 amp., the required current.
(C)
1-0 ul
-0-9
08 07
06
0-6 -0-4
ffl
80 -pO 70-
-01
Fig: 14.
20-
Type B. Nomograms with three rectilinear scales, two of which are parallel. If the formula to be represented can be put in the form
/i +/2 h 3 = o (6)
it can be represented by two systems of points (z\), (zj) arranged along two parallel straight lines, and a third system arranged along a straight line making an angle with the other twe.
As before, we take the functional scales
M=Ml/l
along the parallel axes AM, Bv (fig. 15). We then have for the system (zs)
which with our usual axes defines the system of points
J ' "
so that the points of the system are arranged along AB.
The scale (z 3 ) can be graduated by the use of the above expression for x, or from a double-entry table of cor- responding values of z\, Zi, z>, by suc- cessive alignments of pairs of values of Zi, Z 2 corresponding to any gradu- ation z 3 . Thus if a and c-are a pair of values of z\ and Z2 which correspond to the value b of zs, we join ac to cut AB at b, which gives the graduation of the scale for the value b.
If A and B do not appear on the diagram the support of the scale (z 3 ) can be drawn by making use of the relation, that if 62 are the distances of a point on (z 3 ) from AK, Bt>, we have
Fig: 15.
M2
On the completed diagram any straight line drawn across the three scales will cut them at corresponding values of Zi, Z2, Za as defined by (6).
The scale (z 3 ) will lie between or outside the scales (zj), (z2) according as to whether h 3 is positive or negative, and, as in the previous type, it is as a rule best to arrange that the scale of the variable, which generally has to be determined in terms of the other two, lies between their scales; hs can always be made positive or negative as desired, altering if necessary the signs of both ft and h>.
As an example of Type B take Sir Benjamin Baker's Rule for the weight of rails
W=I7\(L + 0-0001 Lt^)" where L = Greatest load on one driving wheel in tons.
o = Maximum velocity in miles per hour. W = Weight of rails in Ib. per yard.
Writing the formula
and taking
. i +0-000 1 v 1 zi=L, /i=L a=, ft = -,+Q.OOO i V
"=O
construct the scales M
,,-w.*,= Q'
fi=Mi L
>tl \ i+o-oooif 2
