4^8 Mb DE morgan, ON THE THEORY OF 



Here h has two values, which become imaginary when SA^- A^ is negative. We may reject 

 this supposition : for when the first approximation is absolutely true, we have sA^-Ai = 0, and 

 we may presume that, in any law we shall have to represent, large errors are more infrequent 

 than in the first modulus, so that A^, when second approximation is necessary, loses more than 

 A^. The two values of h are greater and less than A.^, so that the values of q have different 

 signs, with values oi p greater and less than unity. To determine which sign q should take, 

 observe that I5p — A^h'^ will, from the third equation, take a different sign from q ; so that 



AJi^ - AJi^ 



2 2 



will have a different sign from q. This quantity vanishes in the first approximation, and 

 will, as above explained, become positive in any law we may have to represent. Let 



A, = sA^\l-a% 



a being positive. This gives 



z , N y Z + Sa ' -a I 



A = (1 + a) Jo , » = 5 q = —, r„ • —r • 



And a = ^(SA^^ — Ai) : A2\/3, by the magnitude of which we judge of the necessity for 

 a second approximation. 



Our modulus is now \/- .e"'*' (p + qx^), and for the chance of an error lying between 



TT 



- m and + m we have 



remembering that p + qh = 1. Now q : 2c being small, and m rather small in all cases in 

 which 6""" is not, Taylor's theorem shows that the preceding is very close to 



* IT Jq - -y/ir •'0 



^ \ 2c J ^y2.Ai y/{\ + a) \ 2 (1 + a)\ y^2A.i \ 8 J 

 nearly. Hence the probable error. is 



•476936 ^/2A2 .(1 +- aA , 



which is that of the first approximation increased in the proportion of 8 to 8 + So*, or 8^2' 

 to 11.^2* - At. This change is small if the first approximation be good; for then At = SA^^ 

 nearly : but if this be not the case, the alteration of the probable error is of importance. 



The reader will observe that since q is negative, our second approximation involves the 

 supposition that, when w is great enough, the modulus is negative, which is incapable of inter- 



