The formula referred is the condition for the coplanarity of the extremities of four coinitial vectors; namely, if α, β, γ, δ, are the vectors, then

(See Kelland and Tait's Quaternions, p. 62.)

In the first place, it may be pointed out that these equations do involve one condition on the four vectors α, β, γ, δ. For, since the vanishing of a vector involves three conditions, and only the three ratios of the four quantities a, b, c, d, are involved, it is always possible to determine these four quantities so that the condition aα + bβ + cγ +dδ = 0 shall be satisfied. But, in general, the equation a + b + c + d = 0 will not be satisfied; and thus one condition is involved.