456 



from (4) 



PRACTICAL MATHEMATICS 



27-5 



b = 



27-5 



0-6982 x 0-3018 

 = - 130-5 



from (1) a = 20-1 + 130-5 x 0-6982 



= 111-2 



The probable law is therefore 



y= 111-2- 130-5 e --898* 



222. Another question arises out of the work of the previous 

 paragraph ; that is : knowing that a certain curve follows one 

 or other of the laws, to find the constants of the law. In general, 

 if a law contains three constants, then three points must be taken 

 on the curve, and, substituting the values of the co-ordinates of 

 these points in the law will give three equations to be solved for 

 the three constants. The solution of these equations depends 

 upon the way the values of x are chosen when taking the points on 

 the curve. 



Case I. For the law y = a + bx n , let the values of x be so chosen 

 that they are in Geometrical Progression ; that is, they increase 

 by a common ratio. 



Let the co-ordinates of the three points be (h, yj, (hr, y 2 ), and 



a+bh n =y 1 ... (1) 



a + bh n r n = y ti ... (2) 



a + bh n r zn = y 3 ... (3) 



bh n (r n -l) = 7/2 - y 1 . . . (4) 



bh n r n (r n -l) = y 3 -y 2 . . . (5) 

 _ 2/ 3 - 2/2 



Then for the first point 



for the second point 

 for the third point 

 subtracting (1) from (2) 

 subtracting (2) from (3) 



dividing (5) by (4) 



2/2-2/1 



Thus giving a relation from which the value of n can be calcu- 

 lated, and knowing n, a and b can be found. 



Example. The curve y a + bx n passes through the three 

 points (3, 21-47), (6, 37-09), and (12, 94-36). Find the values of 

 the constants a, b, and n. 



a+b 8" = 21-47 



a + b 6" - 37-09 



a + b 12" - 94-36 



Then b S n (2 n - 1) = 15-62 



and b 3 n 2 n (2 n - 1) = 57-27 



