## Expressible Sentence Examples

- It will be shown later that all invariants, single or simultaneous, are
**expressible**in terms of symbolic products. - If, however, an amount of energy a is taken up in separating atoms, the ratio is
**expressible**as C p /C„= (5+a)/(3-Fa), which is obviously smaller than 5/3, and decreases with increasing values of a. - Also, as the Cartesian geometry shows, all the relations between points are
**expressible**in terms of geometric quantities. - P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are
**expressible**as rational and integral functions. - Resultant
**Expressible**as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is**expressible**as a determinant, viz. - " The symmetric function (m ï¿½8 m' 2s m ï¿½3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is
**expressible**as a linear function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed." - We cannot, however, say that it is an invariant unless it is
**expressible**in terms of the real coefficients. - When either of the forms is of an order higher than the first (ab), as not being
**expressible**in terms of the actual coefficients of the forms, is not an invariant and has no significance. - It has been shown by Gordan that every symbolic product is
**expressible**as a sum of transvectants. - Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 ï¿½?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; ï¿½'ï¿½1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always
**Expressible**By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2. - Associated Forms.-A system of forms, such that every form appertaining to the binary form is
**expressible**as a rational and integral function of the members of the system, is difficult to obtain. - Similarly regarding 1 x 2 as additional parameters, we see that every covariant is
**expressible**as a rational function of n fixed covariants. - Every covariant is rationally
**expressible**by means of the forms f, u 2, u3,... - Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally
**expressible**in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned. - The Hessian 0 =A 2 is such that (f, 2 and if f is
**expressible**in the form X(p x) 3 +,i(g x) 3, that is as the sum of two perfect cubes,. - -, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are
**expressible**as transvectants of compounds of these four. - On this principle the covariant j is
**expressible**in the form R 2 j =5 3 + BS 2 a+4ACSa 2 + C(3AB -4C)a3 when S, a are the above defined linear forms. - The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is
**expressible**as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5. - 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. - The simplest invariant is S = (abc) (abd) (acd) (bcd) cf degree 4, which for the canonical form of Hesse is m(1 -m 3); its vanishing indicates that the form is
**expressible**as a sum of three cubes. - The binary products e i e j, however, are
**expressible**as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question. - But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is
**expressible**as a quadratic function of the components U, V, W, P, Q, R. - The frequency ratios in the diatonic scale are all
**expressible**either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood. - The view that our knowledge in such cases may be completely represented by means of laws of action at a distance,
**expressible**in terms of the positions (and possibly motions) of the interacting bodies without taking any heed of the intervening space, belongs to modern times. - Although, however, gravitation has formed the most perfect instance of an influence completely
**expressible**, up to the most extreme refinement of accuracy, in terms of laws of direct action across space, yet, as is well known, the author of this ideally simple and perfect theory held the view that it is not possible to conceive of direct mechanical action independent of means of transmission. - Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not
**expressible**in finite terms by means of functions such as algebraical or circular functions. - The area is (b 2 +a 2 /2)7r, and the length is
**expressible**as an elliptic integral. - The area of a lune or meniscus is
**expressible**as the difference or sum of two segments, and the circumference as the sum of two arcs. - A much less wise class than the 7r-computers of modern times are the pseudo-circle-squarers, or circle-squarers technically so called, that is to say, persons who, having obtained by illegitimate means a Euclidean construction for the quadrature or a finitely
**expressible**value for 7r, insist on using faulty reasoning and defective mathematics to establish their assertions. - Good as a true universal can only be realized by a true self, and both imply a principle of unity not wholly
**expressible**in terms of the particulars which it unifies. - Ck is not
**expressible**as the square root of an octic function of 0. - Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be
**expressible**in whole numbers. - Judgment is an assertion of reality, requiring comparison and ideas which render it directly
**expressible**in words (Hobhouse, mainly following Bradley). - The importance of a study of the changes of the vis viva depending on squares of velocities, or what is now called the "kinetic energy" of a system, was recognized in Newton's time, especially by Leibnitz; and it was perceived (at any rate for special cases) that an increase in this quantity in the course of any motion of the system was otherwise
**expressible**by what we now call the "work" done by the forces. - Similarly, if a form in k variables be
**expressible**as a quadratic function of k -1, linear functions X1, X2, ... - The sum of the n th powers of the quantities, is
**expressible**in terms of functions which are symbolized by separations of any partition (n"1n'2n'3...) 1 ! - All symmetric functions are
**expressible**in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.