# Expressible Sentence Examples

It will be shown later that all invariants, single or simultaneous, are

**expressible**in terms of symbolic products.Judgment is an assertion of reality, requiring comparison and ideas which render it directly

**expressible**in words (Hobhouse, mainly following Bradley).It has been shown by Gordan that every symbolic product is

**expressible**as a sum of transvectants.The area is (b 2 +a 2 /2)7r, and the length is

**expressible**as an elliptic integral.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.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.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.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.We cannot, however, say that it is an invariant unless it is

**expressible**in terms of the real coefficients.AdvertisementWhen 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.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,...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.AdvertisementThus 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 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.AdvertisementA 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.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.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.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.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.Advertisement