The theory of compound singularities will be referred to farther on.
The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing.
Birds, p. 353), the avifauna of Madagascar is not entirely composed of such singularities as these.
There are other infinite singularities of detail; but the above are more than sufficient to establish the point.
For example, in the curve for gold-aluminium, ignoring minor singularities, we find two intermediate summits, one at the percentage Au 2 A1, and another at the percentage AuAl 2.