Apkil 21, 1916] 



SCIENCE 



577 



per centimeter. Label the points thus located 

 with corresponding Yalues of u. To finish the 

 construction of the chart, connect each value 

 of f(x) on the axis P with the corresponding 

 value of F{x) on the axis B by means of a 

 straight line of indefinite length, which is 

 labelled with the value of x to which it corre- 

 sponds. In Fig. 1, several of such lines have 

 been drawn and marked with the values x^l, 

 x^2, etc. 



Let it be supposed that the values of Uj v 

 and y in any particular example are known, 

 and that the value of x is to be calculated. 

 Locate the point M on the scale of f(u) (axis 

 R), marked with the given value of u. Locate 

 the point N on the scale of f(v) (axis P) 

 marked with the given value of v. Connect M 

 and N, and note the point of intersection, 0, 

 of this line or its prolongation with the hori- 

 zontal axis H. Locate a point A on the scale 

 of f(y) corresponding to the given value of y. 

 From A, draw a line perpendicular to the line 

 MN, and note where its prolongation inter- 

 cepts the middle axis at B. From B, draw a 

 horizontal line, and from C a vertical line, 

 intersecting at D. It will now be noted that 

 D lies on a certain straight line, which is 

 labelled with the value of x required ; or it lies 

 between two such lines, and the required value 

 of X may be read by interpolation. 



In practical work the chart shown in outline 

 in Fig. 1 would be constructed on cross-section 

 paper. We should need, in addition, a sheet 

 of transparent paper or tracing cloth, having 

 two perpendicular lines ruled on its surface. 

 To solve an equation of the form we have been 

 considering, simply move the transparent sheet 

 back and forth over the chart until one of the 

 two perpendicular lines appears to pass through 

 the given value of u at 21, and the given value 

 of V at N, at the same time that the other per- 

 pendicular appears to pass through the point 

 A corresponding to the given value of y. It 

 will now be easy to note the apparent points 

 of intersection of the perpendiculars with the 

 vertical and horizontal axes at B and C respec- 

 tively; and by following along the vertical 

 and horizontal cross-sectioning from B and C 

 we may locate the point D, and thus determine 



the required value of x, without actually draw- 

 ing any construction lines on the chart itself. 

 As a special example of the use of such a 

 chart, let us consider the calculation . of the 

 maximum temperature obtainable with nat- 

 ural gas burned without excess of air. The 

 equation wiU be of the form ax -\- ix'' = c, 

 where the value of the coeiScients a, b and c 

 depend on the composition of the gas, and the 

 specific heat at various temperatures of the 

 products of combustion. In a particular case, 

 let the equation be 



3.2044 t + 0.00074057 f = 8,203 ealories.i 



The construction of the chart is simplified 

 by the fact that the coefficients of t and t- to 



be considered will always be positive (Fig. 2). 

 In this chart, the middle axis Q is moved to 

 the left imtil it coincides with the left-hand 



1 Eioliard 's ' ' MetaUurgieal Calculations, ' ' Vol. 

 I., p. 41. 



