* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

If a particular point on a sphere is ( arbitrarily ) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them.

If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.

If A and B are antipodal points ( like the North pole and the South pole ), then there are infinitely many shortest paths between them.

If one wants to consider antipodal points as identified, one passes to projective space ( see also projective Hilbert space, for this idea as applied in quantum mechanics ).

If the geographic coordinates ( latitude and longitude ) of a point on the Earth ’ s surface are ( φ, θ ), then the coordinates of the antipodal point are (− φ, θ ± 180 °).

If the points aren't thrown all the way, the Turnout cannot be locked, and in turn, the signal cannot be cleared.

If **ya is the multiple secant of **zg which passes through Af and **yb is the simple secant of **zg which passes through Af, and if Af are the points in which **ya meets **zg, and if Af is the image of Af on the generator **yb, it follows that the image of the line Af is Af.

If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).

If the hand value exceeds 21 points, it busts, and all bets on it are immediately forfeit.

If a bubble touches identically-colored bubbles, forming a group of three or more, those bubbles — as well as any bubbles hanging from them — are removed from the field of play, and points are awarded.

If the thumb points in the direction of the 4th substitutent, the enantiomer is R. Otherwise, it's S.

If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval.

If they are successful, they receive two points, if they are not, then they receive zero points.

If they make what would normally be a field goal, they score one point ; what would normally be a touchdown scores two points ( a " two-point conversion ").

If the ball hits the upright above the cross-bar before going through, it is not considered a dead ball, and the points are scored.

If he didn't meet Pitman, Severin points out that Defoe, upon submitting even a draft of a novel about a castaway to his publisher, would undoubtedly have learned about Pitman's book published by his father, especially since the interesting castaway had previously lodged with them at their former premises.

If the difference in vapor pressure between the two components A and B is large ( generally expressed as the difference in boiling points ), the mixture in the beginning of the distillation is highly enriched in component A, and when component A has distilled off, the boiling liquid is enriched in component B.

If air is heated in a region it will expand in all directions such that the velocity field points outward from that region.

If a dachshund is brindled on a dark coat and has tan points, it will have brindling on the tan points only.

If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical ; it is the canonical dual, and the two together form a canonical dual compound.

* If a circle passing through two of the input points doesn't contain any other of them in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points.

If the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes.

If there is a grain of historical truth behind these legends, it may be that Suizei's bare narrative points to one of the chieftains of four clans of the Unebi region who disputed among themselves in the struggle to achieve hegemeony in the Yamato area.

If it is assumed that social change is not only affected, but in some points of view, directed by the advancement of technology, is it ethical to hold these technologies from certain people?

If any part of the ball reaches any part of the imaginary vertical plane transected by this line while in-bounds and in possession of a player whose team is striving toward that end of the field, this is considered a touchdown and scores six points for the team whose player has advanced the ball to, or recovered the ball in, this position.

If you were a man '' --

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If someone were to drop a match in here, this place would go up like a haystack ''.

If the other pilots were worried, they did not show it.

If it were the enemy, tactically his position was correct.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

If an automobile were approaching him, he would know what was required of him, even though he might not be able to act quickly enough.

If there were only darkness, all would be clear.

If there were only the mess, all would be clear ; ;

If these people were denied a voice ( do they have a moral right to a voice??

If we were creating a wholly new society, we could insist that our social, political, economic and philosophic institutions foster rather than hamper man ; ;

If living Jews were unavailable for study, the Bible was at hand.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If she were not at home, Mama would see to it that a fresh white rose was there.

If the Stearns were not there, grace would be omitted.

If anything, the conservative Democrats were more opposed to Hearst than the Republicans.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.

If there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the Hessian troops sent here willy-nilly by the Hessian Government to fight for England in the 1770's were mercenaries, what shall we call the UN troops sent to the Congo willy-nilly by their governments to fight for the United Nations??

If the UN troops are not mercenaries then the Hessians were not mercenaries either.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.