[681 
280 ON THE DERIVATIVES OF THREE BINARY QUANTICS, 
and farther 
{a + /32] ({aOl}, — 2 {al} {/31}, {/901}$A, V, A") = 0, 
or, what is the same thing, 
A + A = {a + /91} {/90}, 
A / + A" = { „ } {aO} , 
({aOl}, — 2 {al} {/91}, {/901}$A, A', A") = 0. 
And in like manner we have 
fx + fi = {a + /92} . 1, 
/ + /*"={ , }•-!, 
({«01}, -2{al}{/91}, {/301}^, /*', /0 = 0; 
and 
v + v = 0, 
i/ + v' = 0, 
({«01}, — 2 {al} {/91}, {/901}$*/, */, i/') = 0. 
We hence find without difficulty 
A , /a , v = /9. /9 + 1, 2./9+1, +1,= {/901} , 2 {/31}, + 1, 
A', z/ = a . /9 , a - /9 , - 1, = {a0} |/S0}, a - /9 , - 1, 
A", /a", i/" = a . a + 1 , — 2 . a + 1, + 1, = {aOl} , 2 {al}, + 1; 
viz. for verification of the A-equations we have 
/9. /9 + 1 . + a. /9 , that is, a + /9 + 1 . /9, = {a -{- /91} {/90}, 
a./3. + a . a + 1, „ a + l+/9.a, ={ „ } {a0}, 
and 
(a.a + 1, — 2. a+ 1./9+1, /9. /9 + l]£/9. /9 + 1, a./9, a.a + l) = 0, 
that is, 
a.a + l./3./9 + l.-2.a+l./9 + l.a./3. + /3./9 + l.a.a+l=0; 
and similarly the /a- and ^-equations may be verified. 
We have thus for the Z’s the equations 
{a + /82} (1), {a + /91} (2), {a + /90} (3), 
a + /9012} Z = 
(/301} , 
2 (/31) , 
+ 1 
„ }Z' = 
{aO} {,30} , 
a — /3 , 
- 1 
„ } Z” = 
{«01} , 
-2(ol} , 
+ 1 
which include the foregoing expressions for Z and Z". 
We may then take the expressions for the W’s to be 
{a + /934} (1), [a + /914} (2), {a + /303}(3), {a+/901} (4), 
{a + /90123} W = 
} W = 
} W" = 
} W'" = 
{