If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

If I now risk some comparisons with Sons And Lovers let it be clear that I am not comparing the two works or judging their merits ; ;

If nothing else, at least two good songs came out of the project, `` Out Of This World '' and `` June Comes Around Every Year ''.

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.

Somebody, got to be somebody If I don't put my two cents in soon, somebody else will I know they're waitin only for one thing: for the bastards what done it to be nailed.

If we cluster together, the redcoats can make an advantage out of it, but there's not a blessed thing they can do with two or three of us except chase us, and we can outrun them ''.

If you've travelled in Europe a time or two, it is quite certain that you've had that wanting-to-be-alone feeling or that you will get it on your next visit across the Atlantic.

If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

If we try to study T using characteristic values, we are confronted with two problems.

If the distribution of the 71 items were wholly concordant in the two families, the distance would of course be 0.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If a child loses a molar at the age of two, the adjoining teeth may shift toward the empty space, thus narrowing the place intended for the permanent ones and producing a jumble.

If you are not well acquainted with the area in which you wish to locate, or if you are not sure that you and your family will like and make a success of farming, usually you would do better to rent a place for a year or two before you buy.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If we add to these contacts with friendly members the `` contacts with an organization of the church '' ( 11.2 per cent of the cases ), then a substantial two thirds of all recruitment is through friendly contact.

If the Orioles are to break their losing streak within the next two days, it will have to be at the expense of the American League champion New York Yankees, who come in here tomorrow for a night game and a single test Sunday afternoon.

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 we look about the world today, we can see clearly that there are two especially significant factors shaping the future of our civilization: science and religion.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

* If two small categories are weakly equivalent, then they are equivalent.

If the demands of these two sovereigns upon his duty of allegiance come into conflict, those of the United States have the paramount authority in American law ; likewise, those of the foreign land have paramount authority in their legal system.

If there is a mixture of only two types of atoms, not counting impurities, such as a copper-nickel alloy, then it is called a binary alloy.

* 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 tile is placed between two hotel chains of the same size, the individual player who places the tile decides which hotel chain remains on the board and which is acquired.

If two players tie for majority, they will share both shareholder bonuses.

If we insert vertices in random order, it turns out ( by a somewhat intricate proof ) that each insertion will flip, on average, only O ( 1 ) triangles – although sometimes it will flip many more.

If v < sub > n </ sub > is the number of vertices of degree n and D is the maximum degree of any vertex,

If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors.

If the edge length of a regular octahedron is a, the radius of a circumscribed sphere ( one that touches the octahedron at all vertices ) is

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 both planes have the same number of vertices,

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 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 there are n vertices in the graph, then each tree has n-1 edges.

If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions:

If all vertices in a tree are within distance one of a central path subgraph, then the tree is a caterpillar tree.

If all vertices are within distance two of a central path subgraph, then the tree is a lobster.

* If we ignore isolated vertices, which will each be their own component of the minimum spanning forest, V ≤ E + 1, so log V is O ( log E ).

If a mesh covers more pixels in screen space than it has vertices, interpolating colour values from samples of expensive lighting calculations at vertices is less processor intensive than performing the lighting calculation for each pixel as in Phong shading.

If n is even there are n / 2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices.

Putting any of the four vertices in the role of O yields four such identities, but in a sense at most three of them are independent: If the " clockwise " sides of three of them are multiplied and the product is inferred to be equal to the product of the " counterclockwise " sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.

For example, given a graph G =( V, E ), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p < sub > 1 </ sub > from u to w and p < sub > 2 </ sub > from w to v are indeed the shortest paths between the corresponding vertices ( by the simple cut-and-paste argument described in CLRS ).

If P has trilinear coordinates p: q: r, then the vertices L, M, N of the pedal triangle of P are given by

If G is a connected graph with infinitely many vertices such that every vertex has finite degree ( that is, each vertex is adjacent to only finitely many other vertices ) then G contains an infinitely long simple path, that is, a path with no repeated vertices.

) If the three vertices are located at,, and, and the sides opposite these vertices have corresponding lengths,, and, then the incenter is at

If repeated vertices are allowed, it is more often called a closed walk.