Schwere, Elektricität und Magnetismus:385

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

By direct multiplication of equations (1) with each other, and of equations (2) with each other, we obtain six of the type

$(i\cdot i')(i\cdot j')+(j\cdot i')(j\cdot j')+(k\cdot i')(k\cdot j')=0.\,$

(4)

By skew multiplication of equations (1) with each other, we obtain three of the type

$k'=\lbrace (j\cdot i')(k\cdot j')-(k\cdot i')(j\cdot j')\rbrace i+\lbrace (k\cdot i')(i\cdot j')-(i\cdot i')(k\cdot j')\rbrace j\,$ $+\lbrace (i\cdot i')(j\cdot j')-(j\cdot i')(i\cdot j')\rbrace k.\,$

Comparing these three equations with the original three, we obtain nine of the type

$i\cdot k'=(j\cdot i')(k\cdot j')-(k\cdot i')(j\cdot j').\,$

(5)

Finally, if we equate the scalar product of the three right hand members of (1) with that of the three left hand members, we obtain

$(i\cdot i')(j\cdot j)(k\cdot k')+(i\cdot j')(j\cdot k')(k\cdot i')+(i\cdot k')(j\cdot i')(k\cdot j')\,$ $-(k\cdot i')(j\cdot j')(i\cdot k')-(k\cdot j')(j\cdot k')(i\cdot i')-(k\cdot k')(j\cdot i')(i\cdot j')=1.\,$

(6)

     Equations (1) and (2) (if the expressions in the parentheses are supposed replaced by numerical values) represent the linear relations which subsist between one vector of one system and the three vectors of the other system. If we desire to express the similar relations which subsist between two vectors of one system and two of the other, we may take the skew products of equations (1) with equations (2), after transposing all terms in the latter. This will afford nine equations of the type

$(i\cdot j')k'-(i\cdot k')j'=(k\cdot i')j-(j\cdot i')k.\,$

(7)