for which the rates of increase in the original function /V) 

 are most or least rapidly increasing, and hence are those 

 for which f"(a)=0. 



To determine these, we see from (14) that, writing as 

 before P—npx v , it is necessary to solve the cubic equat ion. 

 P 3 +P 2 1 3m+3(p- 1) \ + P 1 3m(ro - l)+3m(p - 1), 



+(p-l)(p-2)}-Hn(»-lXm-2) = Q (21) 



Putting P = « — (m+p-1) equation (14) reduces to 

 z 3 - z(2p 2 -Sp + Smp + D + ip* +2mp 2 -2p 2 +p) = 0... (22) 

 If p= 1, equation (14) becomes 



P 3 +3mP a -3m(«!-l)P-r-ro(iw-l)(«i-2) = 0...(23) 

 and (15) then becomes 



»»--3ms + 2m = (24). 



7. Determination of constants of curve.— To find in the 

 equation y = Ax m e m * 



the constants A, m, n, and p, so as to reproduce a given 

 curve from which, let us suppose, the values of // may be 



power of taking the logarithm of the quantities, a process 

 by means of which products are resolved into sums, and 

 exponential forms into products. Further than this it will 

 be necessary also to utilise an operation which takes 

 account of the fact that if a line be divided in a uniform 

 ratio, its logarithmic equivalent will be divided into equal 

 spaces. Thus if the points on a line, reckoned from the 

 origin, be x t xq, xq 2 , xq\ etc., the logarithmic representa- 

 tion of these is log x, log * + log q, log x + 2 log q, log x + 3 

 log q, etc. The solution is then as follows : — 



lo g !h = log A + m log x r -f nx* <25) 



First to eliminate A, we have from this last equation 

 togft- log tf,»log(ffctoi)-r««m log (a «?)...(2») 



We take however four values of the abscissa?, so that 



