fi02 



CURIOUS NUMERICAL PKOPSSITION PEMONSTHATEK 



is shown, that, if the equation be possible, one of the four 

 iollawinor conditions nonst obtain, viz. 



1st 



ad 



)x — z =r >$'■* 2nd ^ .v — z — 5" 



f X •— y z= r" Cx — y — 1"^ 



<x — z — 71"-" ' &" 4th ^ or — z zz 5" 

 iy-Vzzze ly^-zzzn^-H- 



But at present we shall only consider one of those eases, 

 for example the first, and in our result, by substituting 

 j,«* i,.n iriste;ad of r», r^-^ s" for 5", &c., we shall arrive at all 

 the possible cases. First then, let us ascertain whether the 

 ec|uation jc* — ^":^s" be possible, on the supposition that 



y + z zz e" 



Now from these three equations we derive the three fo.U 

 iowiug, viz.- 



xzzl{e ^ (i" + >•") ) 



yzzl[t-h (^°~r")) 



;. = i{r-(."-r")) 

 And conoCquGntly since' x" — y^ zz s", or a;" ~ ^" -f c", 

 therefore 



or 



(^ -i- 5" + r")" -zz {V — 5'^ — r") + (f" 4- 5" -^ r'^)'^ 

 now 



(<"-f5"__y.")nz=f""+Kf"-"(s"— r"}+nflf"-'M5"^r")"-V&c, 

 (^_^^Z7")"rzi""— nt=""-"(5"— r") + «ar"-'^" {f'—f^ — &c, 



And here, since the sum of the two first expressions is equal 

 to the third, it is evident that the latter subtracted from the 

 suin of the two former is equal to zero. But in adding 

 the two first together, the 2nd, 4th, &c, ternas cancel, 

 ^ud consequently in subtracting the letter from that suin> 



tV\e 



