MISCELLANEOUS PROPOSITIONS. 403 



But it is impossible that this result can be universally 

 true. For since a is always finite we can take t so small 

 that ta shall be as small as we please. And as (f> (t) begins 

 with the value zero, if it varies it must at first increase 



numerically with t\ and therefore \J- cannot be greater 



9 w 



than unity. Hence the result is inadmissible ; and it follows 

 that x cannot vary, or in other words, # is always zero. 



372. The preceding proposition may be extended so as 

 to involve any number of such supposed variables as x ; we 

 will take three for example : Having given that if x, y, and z 

 vary, they must be such functions of the independent vari- 

 able t, that 



dx dy , . , T dz 



= - c a y + cjs, 



where a t , a 2 , a 3 , b lf ...c 3 are quantities, not necessarily con- 

 stant, which are always finite ; and having given that x, y, 

 and z are all zero when t is zero: then it will follow that x, y, 

 and z cannot vary, or, in other words, that x, y, and z are 

 always zero. 



Denote x by <f>(t], y by ty(), and z by %(t}. Then, as in 

 the preceding Article, we have 



= t 

 and therefore if < (t) be not zero we have 



and in like manner we deduce two other similar results. 



But it is impossible that these results can be universally 

 true. For suppose t indefinitely small, and let < (t) be not less 

 than either i/r (t) or ^ (t). Then the first of the three results 

 asserts that unity is equal to an indefinitely small quantity. 

 Hence the results are inadmissible ; and it follows that x, y, 

 and z cannot vary, or, in other words, that x, y, and z are 

 always zero. 



DD2 



