943] 
ON RECIPROCANTS AND DIFFERENTIAL INVARIANTS. 
403 
we see at once that in the first equation the terms contain each of them the factor 
a m’+m-i } an q i n the second equation they contain each of them the factor 
omitting these factors, we find 
7 (ya + n'v) — — (ya' 4- nv') 
m' + m — 1 ’ 
m 
— (ya + nv + v) (m - 1) + (/jl! + v) (/x + n'v + v) 
UV> + V '^ + (p + nv ' + v> ) = pi + v i- 
The first of these gives 
/¿i = (m + m — 1) , (ya + nv) - ~ (/a' + nv )}-, 
[m m ) 
which is the value of ya 2 : substituting this in the second equation, we have 
/ 
v x — —, (/a + nv 4- v) (m — 1) — — (a + nv' + v) 
m m 7 
+ (/a' + v) (ya + n'v + v) — (ya + v) (/x + nv + v) 
— (m'-f- m — 1)4 n'v) — (m' + ni — 1) — (uf + nv). 
m 7 v m 
In the left-hand column, the terms containing (ya -1- n'v) are 
iV , _/ P 
= (ya + wV) z/, 
and there are besides the terms —,v(m — 1) + v (/a'4 v), hence the left-hand column is 
O + n'v) j —, (m — l) + fi' + v'— (m' + m - !)[■, 
= vz/ (?z' + 1) 4- /xv + fi'v ( m — 1 + 1) . 
We have therefore 
= z^z/ (n' + 1) + yav + /xv + l) — vv' (n + 1) — ¡xv + 1^ — /x'v, 
that is, 
Vj = vv in' — n) + —7 (m — 1) v — — (mf — l)v'. 
m m x 7 
To complete the proof, it would be of course necessary to compare the remaining 
terms on the two sides respectively: but in what precedes it is shown that, the form 
being assumed, the expression for the alternant must of necessity be (ya, v, m' + m — 1, n'4n). 
where ya, v have their foregoing values. 
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