along two convenient parallel axes. The scale (L) is an evenly divided scale, and to graduate the scale (v) a series of values of v and /a are calculated
30 40 o_ 60 70 80
/2= 0-961 0-917 0-862 0-800 0-735 0-671 0-610 The support of the scale (W) is the straight line joining the zero of the (L) scale to the zero of the fi scale. This latter zero is at an inconveniently great distance from the top graduation on the (v) scale, but the support can readily be obtained without the actual use of the zeros of the /i and /a scales by the use of the formula
referred to above, or by a cross alignment in the following way : Take W=ioo and work out L for B = 6o and 80. The straight lines joining these two values of L and v will intersect at the point loo on the W scale. Joining this point to the zero of the (L) scale gives the support, and the remainder of the scale can be graduated by taking 11 = 50 say, and working out L for W = 2O, 30, 40, 50. ... Joining 50 on the (v) scale to these values of the (L) scale in turn, will give an intersection on the support for the corresponding graduations of the (W) scale.
The completed diagram is shown in fig. 16. To use it sup- pose, for instance, we require the value of Wfor L = 7 tons, = 70 m./hour. The straight line join- ing 7 on the (L) scale to 70 on the (B) scale (see dotted line, fig. 16) cuts the (W) scale at about .82 lb./yd., the required value.
Type C. Nomograms with two parallel rectilinear scales and one curvilinear scale. If the formula to be represented can be put
mi/hr 80-,
-1 + 3 '-SO
in the form '***
/,g3+/2/3+/3 =0 (7)
it can be represented by two systems of points (zi), (22), arranged along two parallel straight lines, and the third system (z 3 ) arranged along a curve.
As in the preceding types we take functional scales
=M2/2
along the parallel axes AM, Bv (fig. 17). We then have for the system (z 3 )
which with our usual axes denn.es the system of points
and we can determine any number of points on the system (z 3 ) by means of these equations, or by a series of cross alignments. This latter method is especially indicated in cases in which a double-entry table of corresponding values of Zi, z 2 , z s , is al- ready available.
Proceeding by whichever of these ways is most convenient, we can ob- tain the complete scale (z 3 ), tracing the curvilinear support through the points determined.
As before it is advantageous, where the variable z 3 is generally the un- known, for the scale (zj) to lie between the scales (zi), (z 2 ). This will be the case if
h 3 Fig: 17. Ya
is positive, and this can always be arranged, if necessary, changing the signs of both / 2 and h 3 .
Having constructed the scales (zi), (z 2 ), (z 3 ) as described above, any straight line drawn across them will cut the three scales at corresponding values of Zi, 22, 23 as defined by (7).
As an example of Type C take the formula used for the thick- ness of cast-iron pipes in waterworks,
where
< = o-oooi25 P d + 0-15
/ = Thickness of metal in inches.
P = Pressure of water in pounds/inch.
d = Internal diameter of pipe in inches.
Writing this,
0-000125 P - / + 0-15 V d = o and putting
, f
we see that it is of type C. We construct the scales
d, hi=l
along two convenient parallel axes (fig. 18).
We then determine sufficient points on (d), by cross alignments, to draw the curve and graduate the scale.
When P is zero,
t=o-i5 Vli giving us an easily calculated series of alignments for d = 5> IO > 'S--
For the cross alignments it will be convenient to take P = 100, and calculate t for a = 5, 10, 15, ... as before.
Suppose now we wish to know the thickness of a pipe of 3O-in. bore to stand a pressure of 130 Ib./in. Joining 130 on the (P) scale to 30 on the (d) scale, and producing the straight line to cut the (t) scale (see dotted line, fig. 18), we get, at the point of intersection, the required value /= 1-3 in.
7. Graphic Representation of Formulae with more than Three Variables, (i.) Double Alignment Nomograms. Certain types of formulae containing four variables can be dealt with by breaking them up into two or three variable formulae with a common auxiliary variable.
Consider for instance a formula which can be written in the form
(Z 3 )
Introducing an auxiliary variable z 6 we can construct two partial nomograms
/.+/*=Z5 - (9)
/ 3 +/4 = 2 6 - ; - (10) of Type A, having the scale (zs) in common.
Such an arrangement is shown sche- (7 j (Z 4 ) matically in fig. 19, the central line repre- senting the auxiliary scale.
If we take values of Zi, Z2, 23, z< satisfy- ing equation (6), the alignment of Zi with z 2 , and of Zs with z 4 will intersect on the scale (25). The central line need not be graduated as it is only required as a refer- ence line, the nomogram being read in the following way:
Suppose we require the value of 24 for
Fig 19.
Join a on (zi) to b on (22) cutting the ref- erence line at e. Join c on (zs) to e and produce to cut (24) jn d, which will give the corresponding value of Z4.
Such a nomogram from the way in which it is read is termed a Double Alignment Nomogram.
It will be noticed that as the unit of (zs) is the same in both (9) and ( 10) we must have the relationship
1+1=1+1
Ml M2 M3 M4
while the distances of the scales from the reference line (fig. 19) will be given by
51 _ _ Mi ** _ _ Ms
5 2 M2* 84 M4
Hence for the practical construction we graduate any of the three scales, (zi), (z 2 ), (z s ), say, from three conveniently chosen origins on the supports of their scales. We then determine a point on the scale (z 4 ) by means of four values of Zi, 2 2 , z 3 , z t , (a, b, c, d, say) which satisfy equation (8).
The alignment of c and the intersection of the alignment ab with the reference line then determine the point d on the scale (z 4 ), and as we know M4 we can construct the scale (z 4 ) completely.
As an example take the formula for the discharge of gas in pipes,
Q=I350D2 ' 5 VoS where
L =Length of pipe in yards.
D = Diameter of pipe in inches.
H =Head of water in inches equivalent to the pressure.
Q = Quantity of gas discharged in cub. ft. per hour. Writing it
log Q+J log L = 2-5 log D +J log H + const. We put
