ON THE USE OF LOGARITHMIC COORDINATES. 161 



to the transformation x = axi, y=^hy^, where a and h are some deter- 

 minate constants. 



Many curves which occur in physics are translatants with respect to 

 all the quantities which are found in the equation; such curves are 

 particularly suitable for treatment by logarithmic coordinates. 



Construction op an Impedance Chart. 



7. As a second example of a translatant, let us take the equation 



which gives the impedance of a circuit whose resistance is R, self-induction 

 L, and subject to a periodic electro-motive force of frequency n, all 

 expressed in some consistent system of units. 



Regarding n as the independent variable, a chart may readily be 

 constructed giving the value of I for any values of R, L, and n ; that is, 

 we can in a few minutes produce what amounts to a complete table of the 

 impedance of all circuits subject to a periodic electro-motive force of any 

 frequency. 



The equation represents a translatant,' for if R and L become R' and 

 L' respectively, the equation remains unaltered if n and I be also changed 



R'T R' 



into =^^ ,n and - I. The first step is to draw the homologue for arbitrary 

 RLi R 



values of R and L. Let R:= 2 and L = "002, these numbers being such 



as might occur in practice, when L and R are expressed in henrys and 



ohms. 



8. The Resistance Line. — If I depended only on the value of R, this 

 would be represented on the chart by a straight line 1 = 2. The line is 

 marked 'Resistance Line ' (see %. 1). 



9. The Self-induction Line. — If I were a function of L and n only, 

 then the value of I would be given by I = 'Inhn. The homologue of this 

 equation is a straight line, making an angle of 4-5°, with the increasing 

 direction of the axis of n. To draw it in the proper position of the chart 

 some one point on it must be found; thus, if n = 100, I =: 1-256. The 

 line is marked ' Self-induction Line ' on the chart. 



10. The Complete Lmpedance Curve. — The actual impedance is due to 

 both effects, and the curvfe showing the relation of I to the other quanti- 

 ties concerned runs above the resistance and self-induction lines ; it is 

 most distant from either line at their point of intersection, and approaches 

 them asymptotically. The curve very soon, however, becomes practically 

 coincident with the straight lines. The curved portion of the homologue 

 may be drawn by the arithmetical computation of a series of points on the 

 curve; it may, however, be drawn with facility by the following device. 



1 1 . Mechanical Device for Drawing the Curved Portion of the Homo- 

 logue. — For any value of n read off on the chart the value of I due to 

 each effect. (In this case the value due to R is constant. It should be 

 noted that when the separate effects are represented by straight lines it is 

 only necessary to read off values through the range m to 10 n, all otiier 

 values being given at once by appropriately shifting the decimal place.) 



' The homologue of any equation of the form y' = aar'"±5fl!" is a translatant. For 

 if a, hx,7jt become a', h\ '"!"/-,*/ • ar, "* T/?^ . y' , the equation is unaltered. 

 1898. H 



