# Theorem Sentence Examples

- 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. - This is the true standpoint from which the
**theorem**should be regarded. - These laws can be established either by tracing the individual terms in a sum or a product or by means of the general
**theorem**in ï¿½ 52 (vi.). - It has been mentioned in ï¿½ 41 (ix.) that the binomial
**theorem**can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account. - Is approximately equal to -J(27rn).(nle) n; the approximation may be improved by Stirling's
**theorem**log e 2 +log e 3 +... - The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important
**theorem**of expressibility is, directly we exclude unity from the partitions, a**theorem**concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. - +amam Expanding the right-hand side by the exponential
**theorem**, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...' - So that A breaks u p into a sum of determinants, and we also obtain a
**theorem**for the addition of determinants which have rows in common. - From the
**theorem**given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz. - Now by the expansion
**theorem**the determinant becomes (-)1 +2+3+ï¿½.ï¿½+2nB.0 = (- I)n(2n +1) +nC =C. - If we form the product A.D by the
**theorem**for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +ï¿½ï¿½ï¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i. **Theorem**of Symmetry.- The similar
**theorem**for n systems of quantities can be at once written down. - To obtain the corresponding
**theorem**concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically. - In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " ï¿½ï¿½ ' axn and there is a remarkable
**theorem**which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively. - Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a
**theorem**due to Sylvester. - We may, by a well-known
**theorem**, write the result as a coefficient of z w in the expansion of 1 - z n+1. - If we expand the symbolic expression by the multinomial
**theorem**, and remember that any symbolic product ai 1 a2 2 a3 3 ... - Now the symbolic expression of the seminvariant can be expanded by the binomial
**theorem**so as to be exhibited as a sum of products of seminvariants, of lower degrees if alai 0-2a2 +...+crea0 can be broken up into any two portions (alai -1-0-2a2-1-ï¿½ï¿½ï¿½ +asas) +(as+1as +1 +o-8+2as+2+ï¿½ï¿½ï¿½ +ooae), such that Q1 +a2+... - A Similar
**Theorem**Holds In The Case Of Any Number Of Binary Forms, The Mixed Seminvariants Being Derived From The Jacobians Of The Several Pairs Of Forms. If The Seminvariant Be Of Degree 0, 0' In The Coefficients, The Forms Of Orders P, Q Respectively, And The Weight W, The Degree Of The Covariant In The Variables Will Be P0 Qo' 2W =E, An Easy Generalization Of The**Theorem**Connected With A Single Form. - Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor
of great elegance, - among which may be mentioned his**theorems****theorem**that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum. - (2) A
**theorem**relating to the apparent curvature of the geocentric path of a comet. - Legendre, in 1783, extended Maclaurin's
**theorem**concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the**theorem**was equally true for any confocal ellipsoids. - The first formal proof of Lagrange's
**theorem**for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality. - This is important when we come to the binomial
**theorem**(ï¿½ 41, and cf. - (iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general
**theorem**that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation. - ï¿½ 21 (ii.)) is that we do not need the general
**theorem**, and that it is unwise to cultivate the habit of laying down a general law as a justification for an isolated action. - The binomial
**theorem**gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial. - Then, provided a r includes the greatest term, it will be found that (A - a)" lies between 0' r and ar+1ï¿½ For actual calculation it is most convenient to write the
**theorem**in the form methods of procedure. - (ii.) We can prove the
**theorem**of ï¿½ 41 (v.) by a double application of the method. - (c) Thus, if the
**theorem**of ï¿½ 41 (v.) is true for r= p, it is true for r= p+1. - The binomial
**theorem**for positive integral index may then be written (x + y) n = -iyi +. - Application of Binomial
**Theorem**to Rational Integral Functions. - The binomial
**theorem**may, for instance, be stated for (x+a)n alone; the formula for (x-a)" being obtained by writing it as {x+(-)a} n or Ix+(- a) } n, so that (x-a) n =x"- 1)xn-laF...+(-)rn(r)xn-rar+..., where + (-) r means - or + according as r is odd or even. - The argument involves the
**theorem**that, if 0 is a positive quantity less than I, 0 t can be made as small as we please by taking t large enough; this follows from the fact that tlog 0 can be made as large (numerically) as we please. - By means of this
**theorem**it can be shown that, whatever the value of n may be, f 1 + (plq)(i)x+(p/q)(2)x2+... - (viii.) In applying the
**theorem**to concrete cases, conversion of a number into a continued fraction is often useful. - This paper is principally based on the following general
**theorem**, which is a remarkable extension of Pascal's hexagram: "If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c., respectively, then the free summit moves on a curve of the degree 2mnp. .. - In terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr.
**Theorem**of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ... - Hence the
**theorem**of expressibility enunciated above. - There are extensions of the binomial
**theorem**, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly. - The successive factors on the right-hand side of Wallis's
**theorem**it 2.2 4.4 6.6 2 = 1.3. - With Descartes the use of exponents as now employed for denoting the powers of a quantity becomes systematic; and without some such step by which the homogeneity of successive powers is at once recognized, the binomial
**theorem**could scarcely have been detected. - Another curious
**theorem**proposed by Bouilland in 1625 as a substitute for Kepler's second law is that the angular motion of the body as measured around the empty focus F' is (approximately) uniform. - Consideration of the binomial
**theorem**for fractional index, or of the continued fraction representing a surd, or ofsuch as Wallis's**theorems****theorem**(ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e. **Theorem**.- The function symbolized by (n), viz.- Hence the
**theorem**is established. - This is the remainder-
**theorem**; it may be proved by induction. - This is the binomial
**theorem**for a positive integral index. - N1 n 2 n 3 ï¿½ï¿½ ï¿½ and, by the auxiliary
**theorem**, any term XmiXm2X, n3 ... - Hence the produc J1 t
**theorem**(21, Z2,...zn / (y1, Y2,...y.n) = ? - For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's
**theorem**of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. - This
**theorem**is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed. - We therefore have the fundamental
**theorem**that the angular velocity of the body around the centre of attraction varies inversely as the square of its distance, and is therefore at every point proportional to the gravitation of the sun. - +o (m l m2 m3) +...,
**Theorem**of Expressibility. - This
**theorem**was investigated by Sir W. - X n / I l yl, y2,ï¿½ï¿½ï¿½yn
**Theorem**.-If the functions y 1, y2,ï¿½ï¿½ï¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.