# Symbolic Sentence Examples

- There are several
**symbolic**acts descriptive of the siege. - It will be shown later that all invariants, single or simultaneous, are expressible in terms of
**symbolic**products. - Of the
**symbolic**factors of the form are replaced by IA others in which new variables y1, y2 replace the old variables x1, x 2 . - The loculi were intact and the epitaphs still in their places, so that " they form a kind of museum, in which the development, the formulae, and the
**symbolic**figures of Christian epigraphy, from its origin to the end of the 3rd or 4th century, can be notified and contemplated, not in artificial specimens as in the Lateran, but in the genuine and living reality of their original condition." **Symbolic**Identities.- For the purpose of manipulating**symbolic**expressions it is necessary to be in possession of certain simple identities which connect certain**symbolic**products.- Early in 1886 he struck the public taste with precision in his wild
**symbolic**tale of The Strange Case of Dr Jekyll and Mr Hyde. - The identities are, in particular, of service in reducing
**symbolic**products to standard forms. A**symbolical**expression may be always so transformed that the power of any determinant factor (ab) is even. - Such a
**symbolic**product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,... - That the name also prevailed as that of a god among other Semitic races (or even These sacred arks were carried in procession accompanied by
**symbolic**figures. - Constitutional formulae like these, in fact, are nothing more than
**symbolic**expressions of the character of the compounds which they represent, the arrangement of symbols in a certain definite manner being understood to convey certain information with regard to the compounds represented. - We cannot even outline here the process of selection by which the
**symbolic**meanings now stereotyped in the Roman Pontifical were arrived at. - Stroh, from a knowledge of the results, was able to verify and extend the results by the
**symbolic**method. - To find the effect of linear transformation on the
**symbolic**form of quantic we will disuse the coefficients a 111 a 12, a21, a22, and employ A1, Iï¿½1, A2, ï¿½2. - The waratah or native tulip, the magnificent flowering head of which, with the kangaroo, is
**symbolic**of the country, is one of the Proteaceae. - With the amice, "Place on my head the helmet of salvation," &c. For the
**symbolic**meanings of the various vestments see the separate articles devoted to them. - And that thence every
**symbolic**product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x. - To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a
**symbolic**form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 ï¿½ al = a;b4 -}-alb? **Symbolic**Identities.- For the purpose of manipulating**symbolic**expressions it is necessary to be in possession of certain simple identities which connect certain**symbolic**products.- Two methods of treatment have been carried on in parallel lines, the unsymbolic and the
**symbolic**; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter. - A separation is the
**symbolic**representation of a product of monomial symmetric functions. - It is clear from what has been said above that the liturgical vestments possessed originally no mystic
**symbolic**meaning whatever; it was equally certain that, as their origins were forgotten, they would develop such a**symbolic**meaning. - It is thus possible to study simultaneously all the theories which depend upon operations of the group.
**Symbolic**Representation of Symmetric Functions.-Denote the s 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,... - May be the same or different, it is necessary that every product of umbrae which arises in the expansion of the
**symbolic**product be of degree n, in a l, a 2; in the case of b,, b 2 of degree n 2; in the case of c 1, c 2 of degree n3; and so on. - Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, which formed a general
**symbolic**method of logical inference. - -2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the
**symbolical**representation of the same covariant; they are equivalent**symbolic**products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ. - Given any propositions involving any number of terms, Boole showed how, by the purely
**symbolic**treatment of the premises, to draw any conclusion logically contained in those premises. - The repetition of the process brought the same results.
- Every
**symbolic**product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),... - It has been shown by Gordan that every
**symbolic**product is expressible as a sum of transvectants. - Among the
**symbolic**conceits in which the authors of the Brahmanas so freely indulge, there is one overshadowing all others - if indeed they do not all more or less enter into it - which may be considered as the sum and substance of these speculations, and the esoteric doctrine of the sacrifice, involved by the Brahmanical ritualists. - By
**symbolic**means," the definition, as developed by him, is one which would apply to the philosophy of Kant. - Where the multiplications on the leftand right-hand sides of the equation are
**symbolic**and unsymbolic respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities. - If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the
**symbolic**representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 . - 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. - (ab)i(ac)j(bc)k..., that the
**symbolic**product (ab)i(ac)j(bc)k..., possesses the invariant property. - The i t " power of that appertaining to a x and b x multiplied by the j t " power of that appertaining to a x and c x multiplied by &c. If any two of the linear forms, say p x, qx, be supposed identical, any
**symbolic**expression involving the factor (pq) is zero. - For these only will the
**symbolic**product be replaceable by a linear function of products of real coefficients. - And we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The
**symbolic**expression of a covariant is equally simple, because we see at once that since AE, B, Ce,... - = t) 1 v ...axbxcx..., and assert that the
**symbolic**product (ab)i(ac)'(bc)k...aibxc2... - +P=n1, j+k+...+T =n3, It will also be a covariant if the
**symbolic**product be factorizable into portions each of which satisfies these conditions. - If we have a
**symbolic**product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on. - Xy 2 -4z 3 +g2x 2 y+g3x 3, and also the special form axz 2 -4by 3 of the cuspidal cubic. An investigation, by non-
**symbolic**methods, is due to F. - A sort of
**symbolic**retaliation was the punishment of the offending member, seen in the cutting off the hand that struck a father or stole a trust; in cutting off the breast of a wet-nurse who substituted a changeling for the child entrusted to her; in the loss of the tongue that denied father or mother (in the Elamite contracts the same penalty was inflicted for perjury); in the loss of the eye that pried into forbidden secrets. - It consisted essentially in the adoption of Delauny's final numerical expressions for longitude, latitude and parallax, with a
**symbolic**term attached to each number, the value of which was to be determined by substitution in the equations of motion. - In common with other sciences, economics makes use of " abstractions"; but if for some problems we employ
**symbolic**processes of reasoning, we must keep clearly in view the limits of their significance, and neither endow the symbols with attributes they can never possess, nor lose sight of the realities behind them. - = exp,udl where exp denotes (by the rule over exp) that the multiplication of operators is
**symbolic**as in Taylor's theorem. - (1 + a i x) (1+ = s i z we have the
**symbolic**identity +02712+0.3x3+... - The weight of the function is bipartite and consists of the two numbers Ep and Eq; the
**symbolic**expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq. **Symbolic**Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the**symbolic**form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are**symbolical**respreentations of the real coefficients ï¿½o, ai,...**Symbolic**Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the**symbolic**form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are**symbolical**respreentations of the real coefficients ï¿½o, ai,...- We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting (ab) k, (OW for azbx, 4x t ' respectively in the
**symbolic**product subjected to transvection. - Notice, therefore, that the
**symbolic**product (ab)i(ac)j(bc)k... - Some of them imitated the Hebrew prophets in the performance of
**symbolic**acts of denunciation, foretelling or warning, going barefoot, or in sackcloth or undress, and, in a few cases, for brief periods, altogether naked; even women in some cases distinguished themselves by extravagance of conduct. - For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the
**symbolic**forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a? - 1+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+ï¿½ï¿½ï¿½ The most general symmetric function to be considered is E 41 041 8424-3033..ï¿½ .conveniently written in the
**symbolic**form (pigi p2g2 p3go...)ï¿½ Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n. - (multiplication
**symbolic**) ?r1! ?2,ï¿½..