AND OTHER SUED EQUATIONS. 215 



(1) ^ (2) gives, X -f- ^/2ax + 6 zi: ( 1 + «) (aw -fVaV-j-i); 



or, since 1 -j-w =z 0, a; -|- *^2ax-^b nz 0, which verifies (1). 



Secondly, by (IV) >/2ax-{^b=: V {2aVP-f 2a»'' V"'""^ -f-6+4} 



=: V {(«VP-fJ)+2awVa'w=*P+*4-a%'*} 



1 f 1 



(11) ^ (21) gives, X'\-\^2ax 4- J— (l+w^) (a/i'-j-V^'w^+S); 



or since 1 -^ n^ zz 6, x -{- \/2ax -4- 6 zr 0, which verifies (V). 

 1 111 



The preceding solutions may be exhibited thus : — 

 xzzn' (a + V«' + bn'') ... (3), 



tiM X zz n^ (a -{- V«' + *»-'") ... (3')> 

 1 111 



(3) and (3') corresponding respectively to (1) and (1'). 



From (3,) */2ax 4-b — « (a + V«M^^~* ) ... (4) 

 11 1 1 1 1 



„ (3S)V2ffa; ^ b zz nP (a ^ a/o,^ -\- brr'") ... (4') 

 111 1 1 1 



(3) 4- (4), and (3') -f (4'), each give 



X -{- ^2ax -4- b zz 0, 

 i 11 I 



Its it ought to be* 



If now we write — 1 for n, in (1) or (V), and bear in 

 mind that nF zz -— Ij and n* zz 1, we get 



X zza — f/a^ -\- b (5) 



11 11 



But if we make the same substitutions in (3) or (3'), we 



S^* ______ 



xzza-^ Vo* + * (6). 



1 11 I 



(5) and (6) are in fact the values of Xy which we should 



have found if we had solved (a) or (0) by the ordihary 



method. Now we know from principles established in a 

 previous portion of this paper (Art. 3), that no values^ of a, 



other than (5) and (6) can satisfy (a) or (/3.) And, since 



