THE DETERMINATION OF THE CONSTANTS 457 



57-27 



Hence 2 n 



also 



15-62 

 n= 1-875 



15-62 



= 0-747 



and a = 21-47 - b 3 n 



- 15-65 



223. Case II. For the law y = b (x + a} n , the only way in which 

 the value of a can be found is by the graphical solution of an equa- 

 tion, but it will simplify the calculation if very simple values of x 

 are chosen. 



Let the co-ordinates of the three points be (x v y^, (x z> y z ), and 



(* 3 . 2/a)- 

 Then for the first point b (x l + a) n = y l ....... (1) 



for the second point b (x z + a) n = y z . . . . . . . (2) 



for the third point b (x 3 + a) n = y 3 ....... (3) 



dividing (2) by (1) r- ....... (*) 



/#, + a\ Wo 



and n log ( 2 ) = log ** 



3 \BI + a) & y l 



or n{ log (x 2 + a) - log (x l + a) } = log y z - log y l . . . (5) 



dividing (8) by (2) 



or w{log (x 3 + a) - log (a? a + a) } = log ?/ 3 - log y z . . . (6) 



dividing (6) by (5) |S (.+ )" jog (.+ ) ^f !f. ~ ^8 If. 

 log (# 2 + a) - log (#! + a) log w 2 - log y l 



thus giving an equation which must be solved graphically for a. 



Example. The curve y = b (x + a) n passes through the three 

 points (2, 9-49), (6. 30-03), and (10, 59-70). Find the values of the 

 constants a, b, and w. 



b(2 + a) n = 9-49 



b(Q + a) n = 30-03 

 -f a) = 59-70 



(i_Y . 



\2 + a) 



Then U-^J =3-163 



\2 + a/ 



and ( ^) = 1-988 



