[311 
TO EQUATIONS OF THE FIFTH ORDER. 
59 
[311 
n, 
311] TO EQUATIONS OF THE FIFTH ORDER. 59 
which together are 
(V©)» {(l + (~f + y l) (VT) 8 (VT> + (-)“ [(-) 8 1 + (-)r 1] (VT)v (\/T') 8 }, 
an expression which vanishes unless (—) 8 , (—)> are both positive or both negative. The 
forms to be considered are therefore 
i such that the 
(-)“> (~f, (~) y 
+ + + 
- + + 
+ - - 
tave one rational 
The first form is 
(V©)“ {(Vt) 8 (Vt> + (Vtf (Vr)>}, 
3 of the quartic 
which, a, ¡3, 7 being each of them even, is a rational function of ©, T + T', TT'. 
of the quartic 
the same thing, 
The second form is 
<y©)- {(Vf) 8 (Vf)v - (VT) 8 (Vt 7 )?}, 
which, a being odd and (3 and y each of them even, is the product of such a 
function into V© (T — T'). 
0 
oefficients of the 
,tion may be of 
The third form is 
(V©)“ {(VT) 8 (Vt> - (Vf)» (VT') 8 }, 
which, a being even and /3 and y each of them odd, is the product of such a function 
[ations 
into VTf'(T-T'). 
And the fourth form is 
(V©)» {(VT) 8 (VT 7 )? + (Vt> (Vt 7 ) 8 ], 
which, a, /3, y being each of them odd, is the product of such a function into V© (T - T'). 
the cyclical sub- 
■ T'), Ve(T-r), 
Hence if T = p + q T'=p — ^V©> and V© Vp + q V© Vp^-V? V© = s, then 
©, T + T' (= 2p), TT' (=p 2 — ©), VTT' (T - T') (= , 
V© (T - T) (= 2g©), and V© VTT 7 (= s) 
are respectively rational functions. This is the d posteriori verification, that with the 
system of equations 
a = m -f V© + Jp + q V©, &c., V© Jp + q V© Jp — q V© = s, 
any function 
</> («, ¡3, y, + (/3, y, 8, a) + </> (7, 8, a, /3) + 0 (8, cl, ¡3, 7) 
is a rational function. 
8—2