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 the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

If one characteristic distinguishes Boris Godunov, it is the consistency with which every person on the stage -- including the chorus -- comes alive in the music.

If an evil which is certain and extensive and immediate may rarely be compensated for by a problematic, speculative, future good, by the same token not every present, certain, and immediate good ( or lesser evil ) that may have to be done will be outweighed by a problematic, speculative, and future evil.

If engine installation takes 20 minutes, hood installation takes five minutes, and wheel installation takes 10 minutes, then a car can be produced every 35 minutes.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

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.

He once wrote, " If I should go to Poland or Germany, every stone, every tree would remind me of contempt, hatred, murder, of children killed, of mothers burned alive, of human beings asphyxiated.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If there is a correct match for every case, the test suite is said to pass.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If the sets are short to medium length the caller will often try to run the dance until each couple has danced with every other couple both as a 1 and a 2 and returned to where they started.

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

:" If I had stated ... the possibility of the introduction or origination of fresh species being a natural, in contradistinction to a miraculous process, I should have raised a host of prejudices against me, which are unfortunately opposed at every step to any philosopher who attempts to address the public on these mysterious subjects ".

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If every term of every definiens must itself be defined, " where at last should we stop?

* Scanning: If a is the next symbol in the input stream, for every state in S ( k ) of the form ( X → α • a β, j ), add ( X → α a • β, j ) to S ( k + 1 ).

If f: X → Y morphism of pointed spaces, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

If f: X → Y is a continuous map, x < sub > 0 </ sub > ∈ X and y < sub > 0 </ sub > ∈ Y with f ( x < sub > 0 </ sub >) = y < sub > 0 </ sub >, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

If property really does double every seven years then, in 2009, the average Sydney home would have been worth $ 189, 530, 112. 00.

If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b.

If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for the formula above ; after some reworking, we get the following filter representation:

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

If we allow both, the problem is called the quantified Boolean formula problem ( QBF ), which can be shown to be PSPACE-complete.

If the answer is " no ", the formula is unsatisfiable.

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like ( a + b )( a-b )= a²-b² it does not tell us how to think mathematically.

If empirical data reach significance under the appropriate statistical formula, the research hypothesis is supported.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

" If I have learned a formula in gibberish which in any way jogs my memory so as to enable me in each single case to act as though I had a general idea, what possible utility is there in distinguishing between such a gibberish ... and an idea?

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

If ΔW is the amount of work performed during a period of time of duration Δt, the average power P < sub > avg </ sub > over that period is given by the formula

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size | ST | given by the product formula:

If the slope m of a line and a point ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) on the line are both known, then the equation of the line can be found using the point-slope formula:

If the respective impedances of the branches of the hybrid that are connected to the conjugate sides of the hybrid are known, hybrid balance may be computed by the formula for return loss.

If the observer is close to the surface of the earth, then it is valid to disregard h in the term, and the formula becomes

If h is significant with respect to R, as with most satellites, then the approximation made previously is no longer valid, and the exact formula is required:

* If 4a < sub > 0 </ sub > = −, the above formula yields ũ < sup > 2 </ sup > = 0.

* If 4a < sub > 0 </ sub > > −, the above formula yields ũ < sup > 2 </ sup > > 0.