FUNCTIONS OF THE A^TH DEGREE. 239 



The first term in the second member of the equation represents 

 the volume of the block at the rear of the cube 3;^^ Its dimensions 

 are 3;'', y'^ and 3;^ — y^. The second term represents the volume of 

 the block on the right side of the cube 3;^^. The dimensions are 

 y^, y^ and 3/*^ — •3;^ The third term gives the volume of the block 

 covering the upper face of the cube 3/^^. The volume of the three 

 blocks is 3;^^ — 3;^^. This will be the result obtained by adding these 

 three terms together. Each of these three terms may be given a 

 very different interpretation. They are each a difference between 

 two quantities. They are therefore each the difference between the 

 cubes of two other quantities, 



y^y^{y^ — ■3'^) =3'^® — 3;^". 



This result shows that the volume of the block forming the back of 

 the cube whose volume is 3;^^ is also equal to the volume of three 

 blocks between the cubes 3'^^ and y'^\ We may therefore write the 

 value of this term as follows : 



+ / 3/^/3 (3;0_3;-/a) 



_|_yy33;-/3(y_yy3). 



In this case the inner cube would have edges whose length is y^'''^. 

 This length can be laid off on the three axes by purely geometrical 

 means, as has been pointed out in the present paper. 



The second term of Eq. (2), which represents the volume of the 

 block on the right side of Fig. 2, may also represent the volume of 

 three blocks between cubes y^"^ and 3;^". The third term represent- 

 ing the block at the top of the large cube may also represent the 

 volume v^" — 3'^^ These nine blocks would fill the volume occupied 

 by the three outer blocks of Fig. 3. 



The thickness of the three shells filling the volume 3;^^ — •3;^^ is 

 (3;6_3,"/3)^(yy3_y%)_|_(y%_,^5)_ Qf course, all of the terms 

 of Eq. (2) can be similarly treated, and the operation may be re- 

 peated on the terms resulting. 



Equation (2) may also represent the difference between the 

 areas of two squares, y^^ and y^, having sides whose lengths are y^ 

 and y^^^. As the second member has an odd number of terms, the 



