# Substitutions Sentence Examples

- But even when I figure out all the
**substitutions**, deciphering the entire notebook will be a long project. - Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear
**Substitutions**," with the topics determinants, &c., linear**substitutions**, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. - Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear
**Substitutions**," with the topics determinants, &c., linear**substitutions**, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. - Functions, operations, transformations,
**substitutions**, correspondences, are but names for various types of relations. - This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of
**substitutions**; it was shown that there exists an invariant theory connected with every group of linear**substitutions**. - In the theory of surfaces we transform from one set of three rectangular axes to another by the
**substitutions**'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2. - In general in space of n dimensions we have n
**substitutions**similar to X l = a11x1 +a12x2 + ï¿½ ï¿½ ï¿½ + ainxn, and we have to express the n 2 coefficients in terms of Zn(n - I)i independent quantities; which must be possible, because X1+X2+..."IL Xn =xi+x2 +x3 +...+4. - In addition, and transform each pair to a new pair by
**substitutions**, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property. - = (A11+A22)n by the
**substitutions**51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c). - If we solve the equations connecting the original and transformed unbrae we find (A ï¿½) (- a 2) =A i( - A 2) + ï¿½'1A1, (A ï¿½) a1 = A2(- A2)+ï¿½2A1, and we find that, except for the factor (A /), -a 2 and +ai are trans formed to -A 2 and +A i by the same
**substitutions**as x i and x 2 are transformed to i and E2. - Are said to be contragredient when the linear
**substitutions**for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z ï¿½..., Z = A 3 X +ï¿½3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?. - All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of
**substitutions**, of the several sets of variables. - Restricted
**Substitutions**We may regard the factors of a binary n ip equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle. - - The cyclical meaning of the succession of zodiacal signs, though now obscured by interpolations and
**substitutions**, was probably once clear and entire. - The motive of some of the
**substitutions**was to avoid the confusion which must have ensued from the duplication of previously existing native asterisms; thus, the Egyptian and Greek Lions were composed of totally different stars.: Abstractions in other cases replaced concrete objects, with the general result of effacing the distinctive character of the Greek zodiac as a " circle of living things." - His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and
**substitutions**, the theory of functions, differential equations and determinants.