112 Mr. G. K. Winter on the Ratio of maximum 



some to solve numerically. Provided s be small, we may get a 

 very near approximation to the value of the positive root in the 

 following way. 



The value of r will, if s be small, not differ very widely from 

 the value given in equation (2), namely 



V ac\ 



which we will call b. Let 



r=b-\-x, 

 equation (1) then becomes 



(b + x) 4 + s{b + x) 3 -b 4 = 0. 

 Rejecting the second and all higher powers of a?, 

 (& + *)* = ft* + 40**, 

 s(b + x) 3 = + 3sb*x + sb 3 



-b* = -b 4 



\b + x) 4 + s(b + x) 3 -b 4 ={^b 3 + ZsV z )x + sb 3 =ti 



sb s sb 



WTSsb* = ~ Wilis 



x=- 



r=b+x=b— 



4b + Ss 



or 



4 limp 1 



7S?^ *Vjx (4) 



V acX 4 4 //^p 2 + 3 s ' 



an equation easily soluble by the aid of logarithms. 



According to Dr. Matthiessen's experiments, the resistance of 

 a wire of pure annealed copper, one-thousandth of an inch in dia- 

 meter and one foot in length, at 0° Centigrade is 9 - 718 B.A. 

 units. This being the case, if we give p the value of *0005, \ is 

 very easily calculated for any given external resistance, being 

 l"-23482 for each B.A. unit. 



The following Table, giving the value of r as obtained by the 

 numerical solution of equations (2) and (4), will show how nearly 

 the two values approximate ; and the difference between the re- 

 sistance of the coil when wound with the wire of the calculated 

 dimensions and the given external resistance will show how im- 

 portant it is to take the insulating covering into account when 

 determining the size of the wire to use in winding the coil of an 

 electromagnet or galvanometer to produce the best result in a 

 given circuit. 



