Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

For example, a 1901 medical book published for home health use gave the following two " Simple Remedy Formulas " for DYSENTERRY: ( 1 ) Thin boiled starch, 2 ounces ; Laudanum, 20 drops ; " Use as an injection every six to twelve hours "; ( 2 ) Tincture rhubarb, 1 ounce ; Laudanum 4 drachms ; " Dose: One teaspoonful every three hours.

Formulas for addition and multiplication are exactly the same as for the ordinary polynomials, with the only difference that both positive and negative powers of X can be present:

* The Animator's Survival Kit: A Manual of Methods, Principles and Formulas for Computer, Stop-motion, Games and Classical Animators, Faber and Faber, 2002 ( expanded edition 2009, adding ' Internet ' to the subtitle )

Formulas for the other coefficients are much harder to get ; Todd classes of toric varieties, the Riemann – Roch theorem as well as Fourier analysis have been used for this purpose.

Formulas are quite often interpreted as propositions ( as, for instance, in propositional logic ).

Formulas with the form described above are known as 2-CNF formulas ; the " 2 " in this name stands for the number of terms per clause, and " CNF " stands for conjunctive normal form, a type of Boolean expression in the form of a conjunction of disjunctions.

The Animator's Survival Kit: A Manual of Methods, Principles, and Formulas for Classical, Computer, Games, Stop Motion, and Internet Animators.

Formulas for different colors of glass are closely guarded.

Formulas for triose and TP are C < sub > 2 </ sub > H < sub > 3 </ sub > O < sub > 2 </ sub >- CH < sub > 2 </ sub > OH and C < sub > 2 </ sub > H < sub > 3 </ sub > O < sub > 2 </ sub >- CH < sub > 2 </ sub > OPO < sub > 3 </ sub >< sup > 2 -</ sup > + 2H < sup >+</ sup >.

Including but not limited to A Moral Code of Ethics, The Sheet of Fives, Pyramid Formulas for Happiness, Harmony and Health, The Ten Stages of Growth and much more.

* Formulas for fast ( er ) matrix multiplication and inversion.

* Formulas for estimating the height of mountains, dated Bath, 13 February 1817 Annals of Philosophy, 9 ( March, 1817 ), 251 -- 252.

Formulas for Generating Pythagorean Triples

TK Solver is also the platform for engineering applications marketed by UTS, including Advanced Spring Design, Integrated Gear Software, Interactive Roark ’ s Formulas, Heat Transfer on TK, and Dynamics and Vibration Analysis.

* Abramowitz and Stegun, the informal name for Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

Examples of these: ( 1 ) 19th Century American Vice Presidents, ( 2 ) Italian Operas, or ( 3 ) Math Formulas.

Formulas are available to estimate the effective length of a whistle, but an accurate formula to predict sounding frequency would have to incorporate whistle length, scale, gas flow rate, mouth height, and mouth wall area ( see below ).

Formulas exist that can determine the approximate bandwidth of a system given the rise and fall times of the waveform.

* Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions ( with Formulas, Graphs and Mathematical Tables ), U. S. Dept.

His most notable works were his ethnographic studies of the Ghost Dance after Sitting Bull's death in 1890, a widespread 19th-century religious movement among various Native American culture groups, and the Cherokee: The Sacred Formulas of the Cherokees ( 1891 ), and Myths of the Cherokee ( 1900 ), all published by the US Bureau of American Ethnology.

* Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Milton Abramowitz and Irene A. Stegun ( Eds ), National Bureau of Standards, 1964.

The four titles in this edition were: The Arcane Teaching ( hardback ), The Arcane Formulas, or Mental Alchemy ( pamphlet ), The Mystery of Sex, or Sex Polarity ( pamphlet ), and Vril, or Vital Magnetism ( pamphlet ).

This edition consisted of the following titles ( the three titles marked with an asterix (*) are the volumes that had appeared together as The Arcane Teaching in the previous edition ): The One and the Many * ( hardback ), Cosmic Law * ( hardback ), The Psychic Planes * ( hardback ), The Arcane Formulas, or Mental Alchemy ( binding unknown ), The Mystery of Sex, or Sex Polarity ( binding unknown ), and Vril, or Vital Magnetism ( binding unknown ).

A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( Dover: New York, 1972 ), p. 78.

* The Perfect Sausages ( previously The Perfect Sausage and other Fundamental Formulas ), ISBN 0-439-95901-2 ( areas and volumes, ellipsoids and toruses, number formulas ( e. g. triangle, hexagonal ), speed, acceleration, stopping time, distance, force, gravity, projectiles, Money: percentages ; simple and compound interest, permutations and combinations.

Stirling Numbers of the First Kind., § 24. 1. 3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.

Formulas not involving st ( i. e., formulas of the usual language of set theory ) are called internal, other formulas are called external.

Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas ( in the language consisting of and 0 ) into A. Formulas mapped to 1 ( or 0 ) for all A-valuations are called A-tautologies.

Formulas were typed into a separate section, and referred to data through their range, not their physical position in the sheets.

A collection of some of his famous guitar solos from Morbid Angel's sixth album " Formulas Fatal to the Flesh " were put on a B-side record entitled " Love of Lava " which later were included on the second disc of Morbid Angel's 8th album, " Heretic ".

*" Basic Concepts and Formulas in Microscopy: Numerical Aperture " by Michael W. Davidson, Nikon MicroscopyU ( website ).

During this period he started supervising Ph. D. students, such as James Halpern ( Contributions to the Study of the Independence of the Axiom of Choice ) and Edgar Lopez-Escobar ( Infinitely Long Formulas with Countable Quantifier Degrees ).

Formulas p and q are syntactically equivalent if and only if p ↔ q is a theorem, while p and q are semantically equivalent if and only if p ↔ q is true in every model ( that is, p ↔ q is logically valid ).

A Complete Collection of Thermodynamic Formulas ( abstract ).

On September 16, Sam Rayburn will have served as Speaker twice as long as any predecessor and I am proud to join with others in marking this date, and in expressing my esteem for that notable American, Sam Rayburn.

One finds, for example, that a terse and stereotyped verbal expression, seeming at first to be a mere hollow convention, reveals itself over the months of therapy as the vehicle for expressing the most varied and intense feelings, and the most unconventional of meanings.

this is a `` fade '' ending, and what you have here is a downtalking style of speech, expressing something less than conventional respect for mother.

His point is not that mythology may not be used, but that it may no longer be regarded as the only or even the most appropriate conceptuality for expressing the Christian kerygma.

The pastor writes a personal letter to each individual, expressing his joy over the decision, assuring him of a pastoral call at the earliest convenient time, and outlining the plan for membership preparation classes and Membership Sunday.

Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

The posterior parietal cortex is responsible for combining and expressing positional information and relating it to movement.

Dionysius of Halicarnassus exhorts us to " Observe in Alcaeus the sublimity, brevity and sweetness coupled with stern power, his splendid figures, and his clearness which was unimpaired by the dialect ; and above all mark his manner of expressing his sentiments on public affairs ," while Quintilian, after commending Alcaeus for his excellence " in that part of his works where he inveighs against tyrants and contributes to good morals ; in his language he is concise, exalted, careful and often like an orator ;" goes on to add: " but he descended into wantonnness and amours, though better fitted for higher things.

Phillip wrote to Townshend from Rio de Janeiro on 25 April 1783, expressing his disappointment that the ending of the American War had robbed him of the opportunity for naval glory in South America.

A controversial figure, Thackeray has attracted significant attention by making statements expressing admiration for Hitler as an artist and demagogue, while condemning the Holocaust, inciting violence against Muslims, expressing support for the LTTE, and taking strong stances on the aspects of popular culture, including fervent opposition to the celebration of Valentine's Day.

In the field of antennas, two different methods of expressing relative bandwidth are used for narrowband and wideband antennas.

The Catholic Church does recognise as valid ( though illicit ) ordinations done by breakaway Catholic, Old Catholic or Oriental bishops, and groups descended from them ; it also regards as both valid and licit those ordinations done by bishops of the Eastern churches, so long as those receiving the ordination conform to other canonical requirements ( for example, is an adult male ) and an orthodox rite of episcopal ordination, expressing the proper functions and sacramental status of a bishop, is used ; this has given rise to the phenomenon of episcopi vagantes ( for example, clergy of the Independent Catholic groups which claim apostolic succession, though this claim is rejected by both Orthodoxy and Catholicism ).

After expressing his interest to play in the 2011 fourth edition of the Indian Premier League ( IPL ), and despite not having have played active cricket for four years, Brian Lara still managed to attract the highest reserve price of $ 400, 000 ahead of the IPL players ' auction in early January 2011 ; however, no franchise bought him.

are just different notations for expressing an inner product between two elements in a Hilbert space ( or for the first three, in any inner product space ).

References to cannibalizing the enemy has also been seen in poetry written in the Song Dynasty, ( for example, in Man Jiang Hong ) although the cannibalizing is perhaps poetic symbolism, expressing hatred towards the enemy.

Dead Kennedys were noted for the harshness of their lyrics, which generally combined biting social satire while expressing a staunchly left-wing view of contemporary America.

He played golf frequently during and after his presidency and was unreserved in expressing his passion for the game.

The double-slit experiment ( and its variations ), conducted with individual particles, has become a classic thought experiment for its clarity in expressing the central puzzles of quantum mechanics.