CHAP. XIV.] EXAMPLE OF ANALYSIS. 221 



xy + (t + tv + tvz + tvzw) xy + ( v + tzwv) xy + tzwxy = 0; (1) 

 and this is the form of reduction sought. 



2. Now according to the principle asserted in Prop, in., 

 Chap, x., the whole relation connecting any particular set of the 

 symbols in the above equation may be deduced by developing 

 that equation with reference to the particular symbols in question, 

 and retaining in the result only those constituents whose coef- 

 ficients are unity. Thus, if x and y are the symbols chosen, we 

 are immediately conducted to the equation 



## = 0, 

 whence we have 



with the interpretation, If gravitation is necessarily present, mat- 

 ter is not a necessary being-:' 



Let us next seek the relation between x and w. Developing 

 (1) with respect to those symbols, we get 



(y + ty + tvy + tvzy + tvzy) xw + (y + ty + tvy + tvzy) xw 



-f (vy + tzvy + tzy) xiv + vyxw = 0. 



The coefficient of xw, and it alone, reduces to unity. For 

 tvzy + tvzy = tvy, and tvy + tvy = ty, and ~ty -I- ty = y, and lastly, 

 y + y = 1. This is always the mode in which such reductions 

 take place. Hence we get 



xw = 0, 

 .-. to -(!-*), 



of which the interpretation is, If motion exists, matter is not a ne- 

 cessary being. 



If, in like manner, we develop (1) with respect to x and z, 

 we get the equation 



xz = 0, 





 ***** 



with the interpretation, If matter is a necessary being, the world 

 is merely material, and without a presiding intelligence. 



