The contents of such a circular group are symbolically (or analogically) represented by the contents of all other circles in the animal kingdom.
1 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.
- If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -ï¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al °, aol, ...) =f exp(ï¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.
We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.
Ternary and Higher Forms.-The ternary form of order n is represented symbolically by (aixl+a2x2+a3x3)' =a'; and, as usual, b, c, d,...
The Hessian is symbolically (abc) 2 azbzcz = H 3, and for the canonical form (1 +2m 3)xyz-m 2 (x 3 +y 3 +z 3).
The extraordinary advantage of the transformation of S2 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to.
Symbolically, to be the fundamental form of seminvariant of degree 0 and weight w; he observes that every form of this degree and weight is a linear lic expressions.
Franciscus Vieta (Francois Viete) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet.
Since the weights used in conjunction with a balance are really standard masses, the word "weight" may be substituted for the word "mass" in the preceding definitions; and we may symbolically express the relations thus: - If M be the weight of substance occupying a volume V, then the absolute density O = M/V; and if m, m 1 be the weights of the substance and of the standard substance which occupy the same volume, the relative density or specific gravity S = m/m l; or more generally if m i be the weight of a volume v of the substance, and m l the weight of a volume v l of the standard, then S = mv l /m l v.
Their relationship and origin from three tribes are symbolically represented by the twin sisters and the two sets of three brothers.
Each sar, month and hour was represented at once visibly and symbolically by a twelfth part of the " furrow " drawn by the solar Bull across the heavens.
These deities were symbolically represented by three men who went round the house by night.
All Israel is represented symbolically in the twelve stones with which he built the altar; and the water which he poured upon the sacrifice and into the surrounding trench was apparently designed to prevent the suspicion of fraud!
He was symbolically elected to the prophetic office by Elijah some time during the reign of Ahab (1 Kings xix.
The modern name, as above-mentioned, is Merj Ibn `Amir (" the meadow-land of the son of `Amir "); in ancient times it was known as the Valley of Jezreel, of which name Esdraelon is a Greek corruption; and by another name (Har-Magedon) derived from that of the important town of Megiddo - it is referred to symbolically in Rev. xvi.
The angle whose tangent is the coefficient of friction is called thf angle of repose, and is expressed symbolically by 4) tan_if.
To express this symbolically, let dii represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction.
To express this symbolically, let Wi, W2 be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R1, Rf the resistances opposed to the effort overcome by Wi, ~AT2 respectively; E1, Ef the shares of the whole energy E exerted upon Wi, W2 respectively.
With the capture of the capitals, the Persian war was at an end, and the atonement for the expedition of Xerxes was completea truth symbolically expressed in the burning of the palace at Persepolis.
The surname "hammerer" might have been applied to Judas either as a distinctive title pure and simple or symbolically as in the parallel case of Edward Scotorum ` malleus."