941] equation Rr + Ss+Tt+U (s 2 - rt) - V= 0. 361 
I take the oppoitunity of remarking' that the complete system of conditions in 
order that the differential 
Adx + Bdy + Gdz + Ddw 
may be = Md U is as follows : viz. writing 
A, B, C, D = 1, 2, 3, 4; 
dB dC dC _ dA dA dB dA dD dB dD dC dD 
dz dy ’ dx dz ’ dy dx ’ dw dx ’ dw dy ’ dw~^Lz~~‘^’ 
where of course 12 = — 21, &c., and 
123 = 1.23 + 2.31 + 3.12, &c.; 1234 = 1.234 - 2.341 + 3.412 - 4.123, 
is =0 identically; 
1234 = 12.34 + 13.42 + 14.23, 
then the conditions equivalent to three independent conditions are 
234 = 0, 341=0, 412=0, 123 = 0, 1234 = 0. 
In fact, the first four equations are 
2.34 - 3.24 + 4.23 = 0, 
-1.34 . + 3.14 + 4.31 = 0, 
1.24-2.14 . +4.12 = 0, 
-1.23-2.31-3.12 . =0; 
hence, multiplying by 1, 2, 3, 4 respectively and adding, we have the identity 1234=0, 
so that these four are equivalent to three independent equations: and multiplying by 
12 . [i — 31 . v + 14 . p, 
— 12.X, . +23.v + 24.p, 
31. X - 23 . y . +34 . P> 
— 14. X — 24 . fju — 34 . v 
respectively, (where X, y, v, p are arbitrary), we have 
(l.X + 2./*+3.i'+4.p) (23.14 + 31.24 + 12.34) = 0, 
that is, 
the fifth condition. 
23.14 + 31.24 + 12.34= 0, or 1234 = 0, 
C. XIII. 
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