For a brief period each year, the rays of the sun are warm enough to melt some of the snows piled a mile deep at the base of the headwalls, and then the pinnacles glisten in the daytime at high noon, and billions of gallons of water begin their slow seepage under the glaciers and across the rockstrewn hanging valleys on their long, meandering journey to the sea -- running east past the sky-carving massifs of Gurla Mandhata and Kemchenjunga, then turning south and curling down through the jungles of Assam, past the Khasi Hills, and into Bengal, past Sirinjani and Madaripur, until the hard water of the melting snows mingles with the soft drainage of fields and at length fans out to meld with the teeming salt depths of the Bay of Bengal.

But that one should superimpose all these charts, run a pin through the common point, and then scale each planetary deferent larger and smaller ( to keep the epicycles from ' bumping ' ), this is contrary to any intention Ptolemy ever expresses.

And if you bore each other then, heaven help you.

the passage and rhythm of time changed, and when I remember back to what happened then, each event is a separate and frozen incident.

Anita put out the remaining candles with a long snuffer, and in the smell of scented candlewick, the comforting awareness of each other's bodies, the retained pattern of dancers and guests remembered, their minds grew numb and then empty of images.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

She then went over them thoroughly giving each a strenuous test in showmanship.

`` We, the Subscribers, do agree, that as soon as a convenient Number of Persons have subscribed to this, or a similar Writing, We will present a petition to the Hon'ble General Court of the Commonwealth of Massachusetts, praying for an Act incorporating into a Body politic the subscribers to such Writing with Liberty to build such a Bridge, and a Right to demand a Toll equal to that received at Malden Bridge, and on like Terms, and if such an Act shall be obtained, then we severally agree each with the others, that we will hold in the said Bridge the several shares set against our respective Names, the whole into two hundred shares being divided, and that we will pay such sums of Money at such Times and in such Manners, as by the said proposed Corporation, shall be directed and required ''.

`` I have come to talk with you about the future of humor and comedy '', I told him, at which he started slightly, and then made us each a stiff drink, with a trembling hand.

The charged particles then repel each other and are also repelled from the charged surface, which almost invariably bears a negative charge under washing conditions.

It was then distilled at least three times from a trap at -78-degrees to a liquid air trap with only a small middle fraction being retained in each distillation.

The contents of the manifold for liquid phase experiments were then mixed by shaking, redistributed to the reaction tubes, frozen down, and each tube was then sealed off.

After blotting out most of the saline around the sections, a drop of Af was layered over each of the sections, allowed to react for 30 minutes, and then washed with PBS for 15 - 30 minutes.

Note also that if Af, then Af is divisible by the polynomial p, because Af contains each Af as a factor.

It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

When Af for each i, we shall have Af, because the operator Af will then be 0 on the range of Af.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

From this initial rational point, one can then compute an infinite number of v < sub > i </ sub >.

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

Copying the numbers that won the previous lottery draw gives an equal probability, although a rational gambler might attempt to predict other players ' choices and then deliberately avoid these numbers.

1308 ), a Friar Minor Conventual like Saint Bonaventure, argued, on the contrary, that from a rational point of view it was certainly as little derogatory to the merits of Christ to assert that Mary was by him preserved from all taint of sin, as to say that she first contracted it and then was delivered.

The problem of rational behavior in this model then becomes an optimization problem, that is:

( which did not exist in Diophantus's time ), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve ; that other point is a new rational point.

If human beings are rational animals of such-and-such a sort, then the moral virtues are ...( filling in the blanks is the difficult part ).

If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean.

Input is received via hearing, sight, smell, taste, or touch and then analyzed through either rational or irrational thought.

Nozick says no, then asks whether we have reasons not to plug into the machine and concludes that since it does not seem to be rational to plug in, ethical hedonism must be false.

Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma ; this does not alter the set of rational roots and only strengthens the divisibility conditions.

Now any rational root corresponds to a factor of degree 1 in of the polynomial, and its primitive representative is then, assuming that p and q are coprime.

If the equation lacks a constant term a < sub > 0 </ sub >, then 0 is one of the rational roots of the equation.

If the starting number is rational then this process exactly parallels the Euclidean algorithm.

Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the " gaps " between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions.

* If X is the Euclidean space, then the interior of the set of rational numbers is empty.

If there is more than one possible equilibrium at any time then the more interesting implications of the theory of rational expectations do not apply.

Some economists now use the adaptive expectations model, but then complement it with ideas based on the rational expectations theory.

He was then asked for a solution of the difficulty, and began to talk trenchant sense, though private anguish showed through in the vehemence of his manner.

For them only a little more needed to be learned, and then all physical knowledge could be neatly sorted, packaged and put in the inventory to be drawn on for the solution of any human problem.

In criticism of the latter's views, his conclusions were based upon dog lung injection studies in which all of the vascular channels were first filled with a solution under pressure and then were injected with various sized colored particles designed to stop at the arteriolar level.

If we have five problems whose solution we seek in relatively united fashion, then there are twice as many issues which, I judge, sharply divide us, intergroup relations practitioners and lay people.

Another common cure was to soak the feet five or ten minutes in warm water, then to apply a solution of equal parts of soda and common brown soap on a kid bandage overnight.

After cyclotron irradiation, the coating was dissolved with nitric acid, and then precipitated as the hydroxide using concentrated aqueous ammonia solution.

When heated to form a solution and then cooled quickly, these alloys become much softer than normal, during the diffusionless transformation, and then harden as they age.

White crystals of a substance called ambrein can be separated from ambergris by heating raw ambergris in alcohol, then allowing the resulting solution to cool.

The air is passed through a spray of salt solution to dehumidify it, then through a spray of water solution to cool it, then another salt solution to dehumidify it again.

After the irradiation, the coating was dissolved with nitric acid and then precipitated as the hydroxide using concentrated aqueous ammonia solution.

The leaching of CuFeS < sub > 2 </ sub > follows the two stages of being dissolved and then further oxidised, with Cu < sup > 2 +</ sup > ions being left in solution.

Because this complex has no charge, it is no longer attracted to polar water molecules and dissolves in the kerosene, which is then easily separated from the solution.

In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Harold Wilson, then the Leader of the Opposition in the Commons, reiterated his belief that a united Ireland was the only possible solution to Northern Ireland's Troubles.

Fundamentally, then, social disorder can stem from the failure to call things by their proper names, and his solution to this was Zhèngmíng ().

The solution sought is then separated from the remaining six based on physical conditions.

When the condition number is exactly one, then the algorithm may find an approximation of the solution with an arbitrary precision.

A single drop of 10 % potassium hydroxide ( KOH ) solution is then added to the specimen.

Adding and subtracting the last two equations then gives the solution