240 NIPHER— FUNCTIONS OF THE A^TH DEGREE. 



final term is resolved into two terms by considering it to be the dif- 

 ference between two squares having areas 3;* and 3;^. The equation 

 may therefore be written 



+ r(r — /) +/(/— 3'') 



+ 3'°(3'' — 3^') +3'' (3'' — 3'') 



+ 3'' (3'" — 3'") +3'*(3'' — 3'") 



+ 3'" (3^* — 3'") +3''(3'* — 3'"") 



+ 3''(3'' — 3'") +3''(3'' — 3'') 

 + 3;2(y_y/.)+y/.(y_y/.). 



Equation (i) may also represent the difference between the vol- 

 umes 3'^- and 3'" of two cubes having edges whose lengths are 3'* and 

 3;^^^ That equation may be written 



yl2_y2^yY(y^ — y^) -^yY(y^ — y^)-^yY(y^ — y') 



^ ySyZ (^y2 _ y2~^ _^ y3y2 (^yZ _ y2 ^ J^ y2y2 (^y3 _ y2 ^ 



+ y-y-{y-—y ) +30' (y- — y ) +y y iy-—y ) 



+ 3'3' (3' — 3''''0+3' 3''^' ( 3' — 3''^' ) + 3'''''3''''' ( 3' — 3' ''' ) • 



Here the final term (3'^ — 3;") of Eq. (i) has been considered as the 

 difference between the volumes of two cubes having edges the 

 lengths of which are y and y"''^. 



The last equation may be written 



3'"-3'— [(y^)'-(3'=')'] + [(3'^)'-(3'')^] + [(3'^)^-r1 + [3'^-3'^]- 



Of course, this discussion does not in any way modify the phys- 

 ical or geometrical meaning of the units linear foot, square foot or 

 cubic foot. A contractor who should order of dealers in such ma- 

 terial 64 feet of garden-hose, 64 feet of sheet tin and 64 feet of 

 sand or gravel, would not learn that any of his orders had been mis- 

 understood. Possibly they might not be understood if he were to 

 order 2*^ feet of each of these materials. 



