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Mathematics
G. S. Khovanskii
On the Representation of the Equation $f_1+f_2+f_3+f_4+f_5+f_6=0$ by a Nomogram with an Oriented Transparent Overlay in the Form of a Ruler
(Presented by Academician A. A. Dorodnitsyn, 28 I 1961)
For an equation with 6 variables
\[ f_1+f_2+f_3+f_4+f_5+f_6=0, \tag{1} \]
where $f_1$ is an abbreviated notation for the function $f_1(\alpha_1)$, etc., it is possible to construct a convenient and easily adjustable nomogram with an oriented transparent overlay in the form of a ruler. Transform equation (1):
\[ f_1+ \frac{ \dfrac{\mu'\mu''}{\mu'+\mu''} f_3+ \dfrac{\mu'\mu''}{\mu'+\mu''} f_6+\mu' f_4 }{\mu'} = -f_2- \frac{ \dfrac{\mu'\mu''}{\mu'+\mu''} f_3+ \dfrac{\mu'\mu''}{\mu'+\mu''} f_6+\mu'' f_5 }{\mu''}, \tag{2} \]
where $\mu'$ and $\mu''$ are parameters satisfying the condition $\mu'\mu''(\mu'+\mu'')\ne 0$.
Set
\[ f_1+ \frac{ \dfrac{\mu'\mu''}{\mu'+\mu''} f_3+ \dfrac{\mu'\mu''}{\mu'+\mu''} f_6+\mu' f_4 }{\mu'} = -\beta+T_4, \tag{3} \]
\[ -f_2- \frac{ \dfrac{\mu'\mu''}{\mu'+\mu''} f_3+ \dfrac{\mu'\mu''}{\mu'+\mu''} f_6+\mu'' f_5 }{\mu''} = -\gamma+T_5, \tag{4} \]
where $\beta$ and $\gamma$ are auxiliary variables; $T_4$ and $T_5$ are arbitrary functions. Then equation (2) is written as
\[ \beta-T_4=\gamma-T_5. \tag{5} \]
We bring equations (3), (4), and (5) to the nomographable form [1]:
\[ \mu'(-\beta-f_1)-\mu'(f_4-T_4) = \mu''(\gamma-f_2)-\mu''(f_5+T_5) = \]
\[ = \frac{\mu'\mu''}{\mu'+\mu''}f_3+ \frac{\mu'\mu''}{\mu'+\mu''}f_6, \tag{6} \]
\[ \beta-T_4=\gamma-T_5=\delta-T_6, \]
where $\delta$ is an auxiliary variable; $T_6$ is an arbitrary function.
The equations of the elements of the nomogram, after the introduction of the transformation parameters $a_0$, $b_0$, $a'_0$, $b'_0$, $a$, $b$, $c$, $d$, $\mu_y$, $\delta_x=0.5(\mu'-\mu'')$, are given in Table 1.
If we set $T_4=T_5=T_6=0$ and $b'_0=b=d=0$, then the scales $\alpha_4$, $\alpha_5$, and $\alpha_6$ will be located on one straight line, and the transparent overlay will have the form of a ruler. From equations (6) we obtain
\[ \beta=\gamma=\delta= \frac{-\mu'(f_1+f_4)+\mu''(f_2+f_5)}{\mu'+\mu''}. \tag{7} \]
An important feature of the nomogram with an oriented transparent overlay in the form of a ruler for relation (1) is the presence of different moduli of the scales of the variables $\alpha_4$, $\alpha_5$, and $\alpha_6$, with two moduli $\mu'$ and $\mu''$ arbitrary,
Table 1
Fixed plane
| Coordinates | Field \((\alpha_1,\beta)\) | Field \((\alpha_2,\gamma)\) | Field \((\alpha_3,\delta)\) |
|---|---|---|---|
| \(x\) | \(a_0-0.5(\mu'+\mu'')\beta-\mu' f_1\) | \(a_0+a+0.5(\mu'+\mu'')\gamma-\mu'' f_2\) | \(a_0+c+0.5(\mu'-\mu'')\delta+\dfrac{\mu'\mu''}{\mu'+\mu''}\,f_3\) |
| \(y\) | \(b_0+\mu_y\beta\) | \(b_0+b+\mu_y\gamma\) | \(b_0+d+\mu_y\delta\) |
Transparent overlay
| Coordinates | Scale \(\alpha_4\) | Scale \(\alpha_5\) | Scale \(\alpha_6\) |
|---|---|---|---|
| \(x\) | \(a'_0+\mu' f_4-0.5(\mu'+\mu'')T_4\) | \(a'_0+a+\mu'' f_5+0.5(\mu'+\mu'')T_5\) | \(a'_0+d-\dfrac{\mu'\mu''}{\mu'+\mu''}\,f_6+0.5(\mu'-\mu'')T_6\) |
| \(y\) | \(b'_0+\mu_yT_4\) | \(b'_0+b+\mu_yT_5\) | \(b'_0+d+\mu_yT_6\) |
and the third, \(\dfrac{\mu'\mu''}{\mu'+\mu''}\), is determined by the same law as the modulus of the mean scale in nomograms of aligned points with parallel scales. In this case the families of parallel straight lines \(\alpha_1\) and \(\alpha_2\) form equal angles with the family of horizontal straight lines of the auxiliary variable \(\beta\), independently of the values of \(\mu'\) and \(\mu''\). The family of parallel straight lines \(\alpha_3\) always makes with the lines \(\beta\) a more obtuse angle than do the lines \(\alpha_1\) and \(\alpha_2\).
Fig. 1. \(a\)—fixed plane; \(b\)—transparent overlay
Figure 1 gives a nomogram for the formula \(z=\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5\). The ranges of variation of the variables are: \(0\leq \alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5\leq 5\), \(0\leq z\leq 25\). In constructing the nomogram, Table 1 takes
\(f_1=\alpha_1\), \(f_2=\alpha_2\), \(f_3=\alpha_3\), \(f_4=\alpha_4\), \(f_5=\alpha_5\), \(f_6=-z\), \(T_4=T_5=T_6=0\), \(b'_0=b=d=0\), \(\mu'=\mu''=\mu_y=5\) mm, \(a_0=b_0=0\), \(a=80\) mm, \(c=115\) mm. The upper and lower directing straight lines correspond to the limiting values of \(\beta\), namely \(\beta_{\min}=-5\) and \(\beta_{\max}=5\), found from formula (7).
If in equations (3)—(6) and in Table 1 one replaces \(T_4\), \(T_5\), and \(T_6\) by the auxiliary variables \(\bar{\beta}\), \(\bar{\gamma}\), and \(\bar{\delta}\), then on the transparent overlay, instead of scales
will be the families of straight lines \(\alpha_4, \alpha_5\), and \(\alpha_6\), parallel respectively to the straight lines \(\alpha_1, \alpha_2\), and \(\alpha_3\) of the fixed plane. The nomogram will have as its elements only families of parallel straight lines.
Let us also note that, by introducing auxiliary variables and arbitrary functions into equation (1), it can be reduced to a nomographable form in four more ways:
\[ \begin{aligned} 1.\quad &(f_1+f_2+T_{12})-\beta = T_{34}-\gamma = (-f_5-f_6)-0,\\ &T_{12}-\beta = (f_3+f_4)-0 = T_{56}-\delta; \end{aligned} \]
\[ \begin{aligned} 2.\quad &(f_1+f_2+T_{12})-\beta = T_{34}-\gamma = -f_5-f_6,\\ &T_{12}-\beta = (f_3+f_4)-0 = \delta-T_6; \end{aligned} \]
\[ \begin{aligned} 3.\quad &(f_1+f_2+T_{12})-\beta = \gamma-T_4 = -f_5-f_6,\\ &T_{12}-\beta = f_3+f_4 = \delta-T_6; \end{aligned} \]
\[ \begin{aligned} 4.\quad &(\beta+f_1)+(f_2-T_2) = \gamma-T_4 = -f_5-f_6,\\ &\beta-T_2 = f_3+f_4 = \delta-T_6. \end{aligned} \]
However, in none of them can the scales of the transparency be arranged on a single straight line.
Computing Center
Academy of Sciences of the USSR
Received
26 I 1961
References
- G. S. Khovanskii, Nomograms with an Oriented Transparency, Moscow, 1957.