Strong^s Problems. 427 



--■ J for its root ; whence, 



by proceeding as before, we find y = — " "* - ^ ^ . But 



X = - ^ - ■ = (by substituting for y its value) 5— ^^ 



2 i X 1 



Again z = — 75 = (by substituting for x its value) 



V4ca — cWy , 

 -i — p ; hence 



^ \4ca — c'a''/ \4ca — c^a^/ 



_ __ 



(by substituting for y and z their values f) and as this also is 



he— 1 

 to be made a square, assume for its root — . Then we 



'h^' "-« (4^^')'''''-"' (4SI?) +'=(*'-')' • 



from which, by reduction, 



e {Aca-c'a?Y-{a?-\-^c) (4ca-cV) 

 0— 2X e2^4ca-cV)2-(2a + c2)2 

 Hence the values of the required numbers are as follows : 



2 h X— 1 

 z = — — , (in which the value of h is to be found from the 



, , ..V a?+2c , 2a+c2 



last equation,) x =1 : — — -, and y = ! ^. 



4ca~ra^ 4ca — cW 



The numbers a, c, and e, are to be so assumed that x, y, and 



z^ may come out positive. If a = 1, c = 2, and 6 = 2, then 



will X = f , 2/ = |, and z ^ \*, which numbers will be found 



upon trial to satisfy the question. It may also be observed 



that c and a being positive, ca must not exceed 4 ; but the 



form of the above expressions for x, y, z, will be sufficient to 



direct us how o, c, and e, are to be assumed. 



