[Footnote *: Strictly, the prismoidal formula should be used, but it complicates the study unduly, and for practical purposes the above may be taken as the volume.]

The average value of a number of samples is the total metal contents of their respective prismoids, divided by the total tonnage of these prismoids. If we let W, W1, V, V1 etc., represent different samples, we have:—

V(WLD/C) + V1(W1 L1 D1/C) + V2(W2 L2 D2/C) = average value.
WLD/C + W1L1D1/C + W2L2D2/C

This may be reduced to:—

(VWLD) + (V1 W1 L1 D1) + (V2 W2 L2 D2,), etc.
(WLD) + (W1L1D1) + (W2L2D2), etc.

As a matter of fact, samples actually represent the value of the outer shell of the block of ore only, and the continuity of the same values through the block is a geological assumption. From the outer shell, all the values can be taken to penetrate equal distances into the block, and therefore D, D1, D2 may be considered as equal and the equation becomes:—

(VWL) + (V1W1L1) + (V2W2L2), etc.
(WL) + (W1L1) + (W2L2), etc.

The length of the prismoid base L for any given sample will be a distance equal to one-half the sum of the distances to the two adjacent samples. As a matter of practice, samples are usually taken at regular intervals, and the lengths L, L1, L2 becoming thus equal can in such case be eliminated, and the equation becomes:—

(VW) + (V1W1) + (V2W2), etc.
W + W1 + W2, etc.

The name "assay foot" or "foot value" has been given to the relation VW, that is, the assay value multiplied by the width sampled.[*] It is by this method that all samples must be averaged. The same relation obviously can be evolved by using an inch instead of a foot, and in narrow veins the assay inch is generally used.