If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

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

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

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 T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* 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 the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

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

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 you can firmly make the good knight sure to pleasure our Corporation '', Sturley wrote, `` besides that ordinary allowance for your diet you shall have 20 for recompence ''.

If, the editors sometimes, dozed and printed pretentious, New, York-mind, dross, they, also printed, Malraux,, Silone,, Chiaromonte,, Gide, Bellow,, Robert Lowell, Francis Fergusson, Mary McCarthy, Delmore Schwartz, Mailer, Elizabeth Hardwick, Eleanor Clark,, and a host of, other good writers.

If a dancer is good, she suggests purely and superbly the fundamental mechanics of ancestry and progeny -- the continuum of mankind.

If you are unable to sign the request, because of illness or other good cause, another person who stands in close personal or business relationship to you may sign the request on your behalf, stating the reason why you are unable to sign.

If the check is not good and the April 15 or other due date deadline elapses, additions to the tax may be incurred.

If you walk into the ring because it is fun to show your dog, he will feel it and give you a good performance!!

If a lot of rust shows in the drain, use a good flushing cleaner.

If it could be shown that judgments of good and bad were not judgments at all, that they asserted nothing true or false, but merely expressed emotions like `` Hurrah '' or `` Fiddlesticks '', then these wayward judgments would cease from troubling and weary heads could be at rest.

If you have a year-round, full-time job you can't expect to grow much more than your family uses -- unless other members of the family do a good deal of the work or you hire help.

If you are considering a part-time farm where the water must be provided by a well, find out if there is a good well on the farm or the probable cost of having one drilled.

If a red Bordeaux of a good name and year is bitter or acid, or cloying and muddy-tasting, leave it alone for a while.

If you're a good ballplayer, you've got to get mad.

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 a man was good, if he was going to be governor, you felt it and you wanted him to go on forever.

If the good deeds of the person outweigh the bad, the soul is worthy of paradise.

If the bad deeds outweigh the good, the bridge narrows down to the width of a blade-edge, and a horrid hag pulls the soul in her arms, and takes it down to hell with her.

* Indivisibility of certain goods: If a person wants to buy a certain amount of another's goods, but only has for payment one indivisible unit of another good which is worth more than what the person wants to obtain, a barter transaction cannot occur.

If they be led by virtue, and uniformity sought to be given them by the rules of propriety, they will have the sense of the shame, and moreover will become good.

If the decrease during exhalation drops the heart rate below 60 bpm on each breath, this type of bradycardia is usually deemed benign and a sign of good autonomic tone.