α,β,γ,δ = 0;
a,b,c,d
a′,b′,c′,d′
a″,b″,c″,d″

or, as this may be written,

α,β,γ, - δ a,b,c = 0;
a,b,c,d a′,b′,c′
a′,b′,c′,d′ a″,b″,c″
a″,b″,c″,d″

which, considering δ as standing herein for its value αx + βy + γz, is a consequence of the original equations only: we have thus an expression for αx + βy + γz, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of α, β, γ on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by

a,b,c ,
a′,b′,c′
a″,b″,c″

are in the first instance obtained in the forms

1 , 1 , 1 ;
a,b,c,d a,b,c,d a,b,c,d
a′,b′,c′,d′ a′,b′,c′,d′ a′,b′,c′,d′
a″,b″,c″,d″ a″,b″,c″,d″ a″,b″,c″,d″

but these are

= b,c,d , - c,d,a , d,a,b ,
b′,c′,d′ c′,d′,a′ d′,a′,b′
b″,c″,d″ c″,d″,a″ d″,a″,b″

or, what is the same thing,