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681] ON THE DERIVATIVES OF THREE BINARY QUANTICS. 
and we obtain in like manner the equations 
X +X' = {a + /3234} {/301}, 
V+\" ={ „ } { «0 } {/30}, 
X" + \'" = { „ }{a01}, 
({a012}, -3{«12} {/32}, +3 {«2} {/312}, - {/3012}#\, V, X", X"') = 0; 
/a + // ={« + /3134}. 2 {/31}, 
/ + /*" = { „ }. a - ft 
/*" + p!" = { „ }. - 2 {al}, 
({«012}, — 3 {«12} {/32}, + 3 {«2} {,812}, - {/3012}$/x, /,>" /0 = 0; 
v + v = {a + /3034}. 1, 
v + v" = { „ }. - 1, 
v"+v'"={ „ }. 1, 
({«012}, -3{«12}{/32}, + 3 {«2} {/312}, -{£012}$*, A 0 = 0; 
p + p =0, 
P +p" =0, 
p" + p"' = 0, 
({«012}, - 3 {«12} {/32}, + 3 {«2} {/312}, - {/3012}#p, p', p", //")={« + /301234}. 
These give for the Xp" square the values 
{/3012} , 3 {/312} , 3 {/32} , +1, 
{«0} {/301}, 2« - /3. {/31}, « - 2/3 - 2, - 1, 
{«01} {/30}, « - 2/3. {«1}, - 2« + /3 - 2, + 1, 
{«012} ,-3{«12} ,+3{«2} , — 1, 
and so on; the law however of the terms in the intermediate lines is not by any 
means obvious. 
Consider now the binary quantics P, Q, R, of the forms (*$#, y)v, (*$&, y)?, 
(*]£#, y) r ; we have for any, for instance for the fourth, order, the dérivâtes 
P(Q, R)\ (P, (Q, R)% (P, (Q, R)J, (P, (Q, P) 1 ) 3 , (P, QRY; 
and it is required to express 
Q(P, RY and P(P, Q)\ 
each of them as a linear function of these. 
C. X. 
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