CHAP. XIV.] EXAMPLE OF ANALYSIS. 223 



termination of the consequences of any complex hypothesis u 9 

 such as, " The world is merely material, and without any pre- 

 siding intelligence (2-), but motion exists" (w), with reference to 

 any other elements of doubt or speculation involved in the origi- 

 nal premises, such as, " Matter is a necessary being" (#), " Gra- 

 vitation is a necessary quality of matter," (y). We should, for 

 this purpose, connect with the general equation (1) a new 



equation, 



u = wz, 



reduce the system thus formed to a single equation, V- 0, in 

 which V satisfies the condition V(\ - V) = 0, and proceed as 

 above to determine the relation between u, x, and y, and finally u 

 as a developed function of x and y. But it is very much better 

 to adopt the methods of Chapters vm. and ix. 1 shall here 

 simply indicate a few resuks, with the leading steps of their de- 

 duction, and leave their verification to the reader's choice. 



In the problem last mentioned we find, as the relation con- 

 necting x, y, w, and z, 



xw + xwy -f xwyz = 0. 



And if we write u = xy, and then eliminate the symbols x and y 

 by the general problem, Chap, ix., we find 



xu + xyu = 0, 

 whence 1 . _ _ 



. _ 

 Qxy 



wherefore _ . , 



wz = - x with xy = 0. 



Hence, If the world is merely material, and without a presiding 

 intelligence, and at the same time motion exists, matter is not a ne- 

 cessary being. 



Now it has before been shown that if motion exists, matter is 

 not a necessary being, so that the above conclusion tells us even 

 less than we had before ascertained to be (inferentially) true. 

 Nevertheless, that conclusion is the proper and complete answer 

 to the question which was proposed, which was, to determine 

 simply the consequences of a certain complex hypothesis. 



