106 



which equals 



Professors Ayrton and Perry on 



V8»-l/ 



or 



8 2 » + l 



n 2 ( 2n +l) 

 $ 2n + 1-2x8"' 



This expression is always greater than n 2 whether n be 

 positive or negative, and is therefore very great since n is 

 great. Consequently for large positive or negative values of 

 m, n, and p, the error (9) becomes very great, and therefore 

 it can have no maximum. The values of a and b, therefore, 

 found from equating to nought its differential coefficients 

 with respect to a and to b, will give a minimum value of 

 this error. And the error will be a minimum when its 

 logarithm is a minimum. Now taking its logarithm, putting 

 r Q equal to 1, and r x equal to 8, differentiating with regard 

 to a and b respectively, and equating to zero, we arrive at 

 the equations 



<$>(n) -0-857 <£(p)=0, (15) 



where 



£(m)-l-143<£(p)=0, 





. (16) 

 . (17) 



To assist in the solution of these simultaneous equations we 

 calculated the following Table : — 



Table I. 



a. 



*(«)• 



- 3 



03293 



- 2 



04670 



- 1 



0-7029 







1-0397 



1 



1-3764 



2 



1-6124 



3 



1-7502 



4-5 



18474 



10 



1-9794 



100 



2-0694 



And in order to solve these equations we used the following 

 artifice: — A curve was first plotted on squared paper with <j>(a) 



