Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.
Further, the numerator factor establishes that these are not all algebraically independent,, but are connected by a syzygy of degree order 6, 6.
1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).
In such cases a bracketed fraction is appended to the specific gravity, of which the numerator and denominator are respectively the temperatures of the substance and of the standard; thus 1.093 (0 0 14°) means that the ratio of the weight of a definite volume of a substance at o to the weight of the same volume of water 4° is I 093.
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.
It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (r/r)th of the logarithm of the quantity.
In refutation of Duchesne(Van der Eycke), he showed that the ratio was 3-, %-, and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 3 6 3 - or ??
We Have Then (26 1 30 (N 6)) R But The Numerator 3 Of This Fraction Becomes By Reduction I I N 40 Or 1 I N To (The 30 Being Rejected, As The Remainder Only Is Sought) =N Io(N I); Therefore, Ultimately, Must Be Deducted From J.
If this arrangement is expressed by a fraction, the numerator of which indicates the number of turns, and the denominator the number of internodes in the spiral cycle, the fraction will be found to represent the angle of divergence of the consecutive leaves on the axis.
With the numerator unity: in order to express such an idea as ~ the Egyptians were obliged to reduce it to a series of primary fractions through double fractions 1~+~1rt~r+1~w+ 1~ 4(1+
Must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive.