FUNCTIONS OF THE A^TH DEGREE. 



237 



lengths 3'i and 3'A can then be laid off. By means of two rulers 

 in parallel, the length 3;^ can similarly be laid off upon both axes. 

 The lengths f^-, f'", etc., can then be similarly laid off upon the two 

 axes. Since 



y xy x^-y x^y 

 I X x^ x^ 



it is evident that any value a""3'"' can be laid oft" by purely geometrical 

 methods. The exponents n and m may be in decimal form and may 

 range in numerical value from o.i to 100 if the experimenter so 

 desires, and all of these values may be represented by lines. 



1 y'^^i y \ y^ y~ 

 Fig. I. 



y- y^ y^ y y' 

 Fig. 2. 



In Fig. 2 we have a system of squares constructed on lines deter- 

 mined by the method described above. The areas of the squares 

 vary from y- to 3;^-. The difference between the areas of the outer 

 and the inner squares is 3/^" — v". 



This value is divisible by y~ — 3'. The value so determined is 



3' — r 



= y'' + y' + y' + y+ y' + y' + y' ^y' + f- + y- 



y-~y 



Multiplying both members of this equation by 3'- — 3', it may be put 

 into the form 



4- 3,5 ( 3,5 — y) _|_ 3,4(3,3 _ 3,4) 



