Fry — Ileal and Complex Numbers as Adjectives or Operators. 27 



OP in the positive direction through a right angle about 0. When OA, 



OB are not taken at right angles, the effect of multiplying (x + iy) by i is 



not so simply represented, nor have we a simple geometrical representation 



of the argument and modulus of ,r +■ iy, nor is the representation of 



(x + iy) (x' + iy') so simply related to the representation of x + iy and 



x' + iy'. 



- , • • • ,'"■-■-, 



By (r+ ty)" ; where m and ?! are positive integers, and — is m its lowest 



terms, we mean a number % such that z" = (x + iy)'". Writing x + iy in the 

 form r (cos + i sin 0), we see that z has the value 



in . . m \ 



cos — + i sin — , 

 n n J 



and ;i - 1 other values obtained by replacing 8 by 



0+2-, + 4tt, . . . , 6 + 2 - 1) - ; 



that these are all different is easily seen from their representative points, 



Ml 



which form a regular polygon of n sides inscribed in the circle of radius = r". 



m 



(x + iy)' » is defined as 



1 -2/ m . . m \ 



= = r n cos — - i sin — o I ; 



2 \ n n J 



{x + iy)" 



and so its n values are obtained by replacing m by - m in the n values of 



m 



(x + iy)". 



As another example of a problem, take the following : — The fore-wheel of 

 a carriage makes 64 revolutions more than the hind-wheel in travelling one 

 mile; but if the circumference of the fore- wheel be increased by 11 inches, 

 it will make only 40 revolutions more than the hind-wheel. Find the 

 circumference of each wheel. With the numbers given, this problem has 

 two real solutions. To understand the negative solution, and complex 

 solutions when the numbers are altered, it is better to alter the problem 

 to the following equivalent one : — There are two lines, P and Q, of which P 

 is contained in a mile 64 times more than Q\ but if the length of P is 

 increased by 11 inches, it is contained in a mile only 40 times more than Q. 

 Find the two lengths. 



Laying down a direction OA, let a be an inch measured in that directum, 

 ft one in the opposite direction. Let xa be the length of P, ya the length of Q, 



