(/6)*{0 + (— PY T/T)" + (—)"U(— 
to Equations of the Fifth Order. 261 
( /o)( VT) V0), 
(-V6)"( VP)"(-VT)’, 
( Ye)(-V/T)*(-V TY, 
(—/6)"(-/T)"( 1)”, 
which together are 
an expression which vanishes unless (—)*, (—)” are both posi- 
tive or both negative. The forms to be considered are therefore 
a A Wa aed eae So 
+ + + 
~ + + 
a — — 
The first form is 
(V/O)* (STP T+ (TPT), 
which, «, 8, y bemg each of them even, is a rational function 
of ©, T+T', TY’. 
The second form is 
(VOUT TY WS TPS TY» 
which, « being odd and £ and y each of them even, is the pro- 
duct of such a function into \/O(T—T’). 
The third form is 
(VOUS TPS TY —(S TS T)*5, 
which, « being even and @ and y each of them odd, is the pro- 
duct of such a function into »/ TT’ (T—T’). 
And the fourth form is 
(VO) US TPS TY + / TS T)*S, 
which, a, 8, y being each of them odd, is the product of such a 
function into »/@(T— 1’). 
Hence if T=p+ q/ 0, T’=p—q/9, and 
/OrV/ptqV/@ /p—q/ O=s, 
®, T+T'(=2p), TY (=p?—¢?20), /TY(T—T' )(= = 
/O(T—T’)(=2¢8), and /O/TT(=s) 
are respectively rational functions. Thisis the @ posteriori veri- 
PL +(-PI (VT) TY} > 
