238 



NIPHER— GRAPHICAL REPRESENTATION OF 



-\-yHy^ — y-) -\-y^(y^ — y^) 

 + 3''(y— 3' ) +3' (y^'—y )• 



(I) 



The first two terms of the second member of this equation rep- 

 resent the area of the two outer rectangles between the two squares 

 having areas y'^^ and y'^°. These rectangles each have a width y^ 



— 3'^, and their combined length is 

 ^6 __!_ ^,5_ -pj^g remaining terms 



represent the remaining strip areas, 

 between the squares 3;^" and y^, 



„18. 



/ 



y y'~ y'^ y* y^ y^ 



y 



Fig. 3 represents a cube, the 

 volume within the outer surface of 

 which is y^^. In the lower left- 

 hand corner is a cube having a vol- 

 ume y^. Between these cubes are 

 fifteen blocks filling the space be- 

 tween the two cubes. If this cube 



were to be placed upon the area forming Fig. 2, each block which 



stands on edge would cover a rectangular area shown in Fig. 2. 



Dividing the difference between the volumes by 3;- — y as in the 



former case we have 



-2/ 

 Fig. 3. 



r 



r 



, yie -^ ys -^ 3,2 



f 



y —y 



Multiplying by y^ — y as before the resulting terms may be written 

 .3;3 = 3,63,6(^,6 _y) ^ 3,63,5 Q,6_y 5) _^yY-(y^ — yr') 

 _j_ 3,53,5 (^5 _y) _|_ 3,53,4(3,5 _y) _|_ 3,43,4(3,5 _y) 

 _j_ 3,43,4 (3,4 _y) _^3,4y(y_3,3) j^yY(y* — y^) 

 + yy(y — 3") -{-y^y-iy^' — y-) -\-y-y-(y^ — y-) 



-{-y-y~(y'—y ) -\-y-y (y-—y ) +y y iy-—y )• (2) 



