578 



SCIENCE 



[N. S. Vol. XLni. No. 1112 



axis, for the reason that the values of y 

 (calories) to be considered will always be 

 negative when transferred to the left-hand 

 side of the equation, and will therefore lie to 

 the right of Q. Since negative temperatures 

 are not to be considered, the horizontal axis 

 may be placed at the bottom of the chart. 



In the construction of the chart let the ver- 

 tical axes be taken 20 centimeters apart. 

 Along the left-hand axis lay off values of t in 

 an upward direction, at the rate of 100 units 

 per centimeter (A = 100) ; in this way, the 

 chart can be used for temperatures up to 

 about 2500° C, if it be 30 centimeters high. 



In a similar way, calculate the values of x- 

 and lay them off in an upward direction along 

 the right-hand axis, at the rate of 100,000 units 

 per centimeter (5^100,000). Since the 

 maximum value of x to be considered is about 

 2,000, the graduations along the right-hand 

 axis will extend about 2,000-/100,000 = 40 

 centimeters above the horizontal axis. 



From left to right along the horizontal axis, 

 lay down a scale for the various values of y, 

 at the rate of 500 units per centimeter 

 (C = 500). In this way, the maximum value 

 of y (about 10,000) will lie about 20 centi- 

 meters from the left-hand end of the hori- 

 zontal scale. 



Along the left-hand axis, in a downward 

 direction from a second horizontal axis 

 (located at any convenient distance above the 

 bottom of the chart), lay down a scale of 

 coefficients of t^, at the rate of 



500 



mB 20 X 100,000 



= 0.00025 units per centimeter. 



Along the right-hand axis, lay off in an up- 

 ward direction a scale of coefficients of t, at 

 the rate of 



C 

 mA 



__500 



20 X 100 



= 0.25 units per centimeter. 



To solve the particular quadratic equation 

 given above, lay a transparent sheet bearing 

 two perpendicular lines over the chart, so that 

 the value of a (3.20), at Mj the value of h 

 (0.00074), at N, and the value of c (8203), at 

 A, are crossed by the perpendicular lines of the 

 transparent sheet. Note the point of intersec- 



tion with the left-hand vertical axis at B, and 

 with the auxiliary horizontal axis at C. From 

 B and follow along the horizontal and ver- 

 tical cross-sectioning (not shown in Fig. 2) to 

 locate the point Z>, where the required value 

 of X (1805°) is read directly from the chart. 

 In Fig. 3 we have a further illustration of 

 the use of such a chart in the solution of the 

 equation 



a log X + b \/x = c. 



There are two values of \/x^ a positive and a 

 negative one, for each value of x or log x. 

 There are accordingly two lines to be drawn 

 from each value of log x on the left axis to 

 connect with the corresponding values of \/x 

 on the right axis. One of the two sets of lines 

 thus formed has been shown in the figure by 

 dashes. 



Solution of particular equation 



72.5 log a; -f 6.25 V'i = 54 



is shown in the chart, the point D representing 

 the value of x required. It will be noticed 

 that in this case there are three different sets 

 of lines that cross the region in which D hap- 

 pens to fall. There are accordingly three real 

 roots to the given equation, the values read 

 from the chart being 3.8, 10.5 and 530. 



It is apparent that the number of real 

 roots to any equation of the general form 

 a log X -\- 'b\/x = c will depend upon the values 

 of the coefficients a, h and c. In the region 

 in which the point D is shown, there are al- 

 ways three real roots, one of these satisfying 

 the equation if a positive value of \/x be 

 taken; the other two if a negative value of 

 \/x be taken. In the region of the chart in 

 which the point B falls, there is but one real 

 root of the equation. If negative values of 

 the coefficient h are considered the chart may 

 need to be extended to the left of the left-hand 

 axis ; there will be two real roots in this region. 

 If negative values of the coefficient a are con- 

 sidered the chart may need to be extended to 

 the right of the right-hand axis. The trend 

 of the lines in the diagram indicates that in 

 this region there will in general be but one 

 real root of the equation when a is negative; 

 but in certain special instances, as for example. 



