An important reference is " The Differential Equations satisfied by Concomitants of Quantics," by A.
A great number of birds' bones have been found in caves, and among them some bearing marks of human workmanship. In France we have a large and extinct crane, Grus primigenia, but more interesting are the numerous relics of two species, the concomitants even now of the reindeer, which were abundant in that country at the period when this beast flourished there,and have followed it in its northward retreat.
The dissolution of feudalism, the development of towns, the growth of scholasticism, all these and much more have been ascribed to the Crusades, when in truth they were concomitants rather than results, or at any rate, if in part the results of the Crusades, were in far larger part the results of other things.
There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.
He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
V., established the important result that in the case of a form in n variables, the concomitants of the form, or of a system of such forms, involve in the aggregate n-1 classes of aa =5135 4 +4B8 3 p) =0, =5(135 4 - 4A 2 p 4) =0, P yield by elimination of S and p the discriminant D =64B-A2.
Hence many structures which are obvious to the eye, and serve as distinguishing marks of separate species, are really not themselves of value or use, but are the necessary concomitants of less obvious and even altogether obscure qualities, which are the real characters upon which selection is acting.
Partly in consequence of this, the deification of the king, with all its concomitants, was gradually extended through the ranks of the noble and wealthy until it came within the reach of the humblest, and even animals shared the honour of deification after death.
Especially Isaiah), the character of the reforms ascribed to Josiah (2 Kings xxiii.), the pictures drawn by Jeremiah and Ezekiel, and the latter's condemnation of the half-Hittite, half-Amorite capital, combine with the events of later history to prove that the religion of the national sanctuary must not be too narrowly estimated from the denunciations of more spiritual minds or from a priori views of the inevitable concomitants of either henotheism or monotheism or of a lofty ethical teaching.
The concomitants of these have been irresponsibility and inefficiency.
But then having thought on a tender way of polishing, proper for metall, whereby, as I imagined, the figure also would be corrected to the last; I began to try, what might be effected in this kind, and by degrees so far perfected an Instrument (in the essential parts of it like that I sent to London), by which I could discern Jupiters 4 Concomitants, and shewed them divers times to two others of my acquaintance.