If the Union conceded this to them, the same right must be conceded to each remaining state whenever it saw fit to secede: This would destroy the federal balance between it and the states, and in the end sacrifice to the sovereignty of the states all the liberty the citizens had gained by their Union.

If the railroads, for example, regularly slaughtered 25,000 passengers each year, the high priests of the cult would have cause to tremble for their personal safety, for such a holocaust would excite demands for the hanging of every railroad president in the United States.

If a branch extended out too far, each man held it back for the next, and if they met a low overhang, each warned the other.

If coffee is sold at the cafeteria, let a few workers in each department get it for the whole group.

If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.

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.

If each text form is marked when matched with a dictionary form, the text forms not contained in the dictionary can be identified when all dictionary forms have been read.

If cell Af is an information cell, it and any information cells in the Y-region that have been linked to Af each contain an address in the W-region where a potentially matching form is stored.

If the bottom name in each column did not have a responsible executive identified, the next name above which identified such a responsible executive was substituted.

If the wedding party lasted late, and the travel schedule means there are only a few hours before resuming the trip or making an early start, the husband may forestall tensions and uncertainties by confiding to his bride that lying in each other's arms will be bliss enough for these few hours.

If he is not told which of four or five readings was meant for him, he can more readily assess each item in a larger frame: `` Does that statement really sound as if it were for me, significant in my particular life??

If laborers are merely commodities competing against each other in a market place like so many bags of wheat and corn ( unsupported, by the way, by any agricultural subsidy ), then they may be pardoned for reacting with complete antagonism to a system that imposes such status upon them.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles.

If the square mile is divided into quarters, each quarter has a side length of mile ( 880 yards ) and is square mile in area, or 160 acres.

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.

If a water absorbing swale for each yard is combined with permeable concrete streets, storm drains can be omitted from the neighbourhood.

If the candidate relies on sponsors, the celebrant asks them if they will raise the child in " the Christian faith and life " ( ECUSA BCP ), and will raise the child through " prayers and witness to grow into the full stature of Christ " to which the parents will state to each, " I will, with God's help.

If the dealer does not bust, each remaining bet wins if its hand is higher than the dealer's, and loses if it is lower.

If all four players pass in the first round, the deal is not played ; in rubber bridge the deal is not scored and the hand is redealt by the original dealer, while in duplicate the score is recorded as zero for each pair and returned to the board.

If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If a nation wished to get a head start in physical fitness over all other nations, it would start its kindergarten students on a program of gymnastics the day they entered and thus eliminate a large number of the problems that plague American schools.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If a wedge-shaped coating of increasing thickness is removed from a substrate by an instrument like the Hesiometer with a knife of constant rake angle, a number of removal mechanisms are often observed which depend upon the thickness of the coating.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If Rock shows up during the number -- or you hear anything -- give me the signal ''.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If so, it might be worth while to assign a future jazz show to a different department -- one with enough confidence in the musical material to cut down on the number of performers and give them a little room to display their talents.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If its minimal polynomial has degree, then the algebraic number is said to be of degree.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.

If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?

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.

If the two carbon atoms at the double bond are linked to a different number of hydrogen atoms, the halogen is found preferentially at the carbon with fewer hydrogen substituents ( Markovnikov's rule ).

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If there is no number to its left, simply look at the column headed " 1 " in the previous row.

If the game has been scored correctly, the total number of plate appearances for a team should equal the total of that team's runs, men left on base, and men put out.

If the other player also knows to offer sacrifices, the expert also has to manipulate the number of available sacrifices through earlier play.