For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

For instance, retaining the first two terms of the series yields the second-order approximation to f ’( x ) mentioned at the end of the section Higher-order differences.

For example, the sentence involving Napoleon can be rewritten as “ any group of people that includes me and the parents of each person in the group must also include Napoleon ,” which is easily interpreted as a statement in second-order logic ( one would naturally start by assigning a name, such as G, to the group of people under consideration ).

For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles ( so power decreases by 12 dB per octave, or 40 dB per decade ).

For an infinitesimal deformation the displacements and the displacement gradients are small compared to unity, i. e., and, allowing for the geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor, i. e. the non-linear or second-order terms of the finite strain tensor can be neglected.

For example, money satisfies no biological or psychological needs, but a pay check appears to reduce drive through second-order conditioning.

For example, while the analysis of a table ( matrix, or second-order arry ) of data is routine in several fields, multiway methods are applied to data sets that involve 3rd, 4th, or higher-orders.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

For example, the second-order sentence says that for every set P of individuals and every individual x, either x is in P or it is not ( this is the principle of bivalence ).

For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀ x ∃ y ( x + y = 0 ) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.

For example, the second-order derivative would be:

For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a of.

For instance, a second-order low-pass notch filter section only reduces ( rather than eliminates ) very high frequencies, but has a steep response falling to zero at a specific frequency ( the so-called notch frequency ).

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows.

For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.

" It is possible to define a formula True ( n ) whose extension is T *, but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic.

For a second-order linear autonomous systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.

For comparison, in the equivalent Euler – Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear ; however, those equations are a system of N, generally second-order equations for the time evolution of the generalized coordinates.

For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic.

For a second-order reaction half-lives progressively double.

* For a second-order polynomial,, all the roots are in the left half plane ( and the system with characteristic equation is stable ) if all the coefficients satisfy.

For example, an animal might first learn to associate a bell with food ( first-order conditioning ), but then learn to associate a light with the bell ( second-order conditioning ).

For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.

For example, among the Saxons of the eighth century social divisions were cast-iron, and the law punished with death the man who should presume to marry a woman of rank higher than his own.

For manga, the most typical divisions are:

For Stephen T. Engel, on the other hand, Rousseau's nationalism anticipated modern theories of " imagined communities " that transcend social and religious divisions within states.

For example, U. S. armored divisions had a balance of three battalions each of tanks, armored infantry, and self-propelled artillery.

For the invasion of France and the Low Countries in 1940 ( Fall Gelb ) the three SS-VT and three of the SS-TV regiments were each organized into divisions ( the future 2nd " Das Reich " and 3rd " Totenkopf "), and another division was raised from the Ordnungspolizei ( later the 4th " Polizei ").

For instance, the Danish Vikings were aware of the internal divisions within Charlemagne's empire that began in the 830s and resulted in schism.

For the full Julian Date ( divisions are real numbers ):

For the execution of these divisions tonguing reached a high degree of complexity.

For the fastest divisions the liquid lingua riversa, usually expressed as le-re-le-re, de-re-le-re te-re-le-re, was prescribed.

For example in the lower divisions wrestlers with the same record in a tournament are generally matched up with each other, and the last matchups often involve undefeated wrestlers competing against each other, even if they are from opposite ends of the division.

For dialects and subdialects, consult the separate main articles for each of the three divisions.

For administrative purposes the Maldives government organized these atolls into nineteen administrative divisions.

For the first time all four Canadian divisions were assembled to participate in a battle.

For administrative purposes, the organization divides itself geographically into territories, which are then sub-divided into divisions.

For a full discussion, see administrative divisions of France.

For example, in Galpin's 1937 book A Textbook of European Musical Instruments, he lists electrophones with three second-level divisions for sound generation (" by oscillation ," " electro-magnetic ," and " electro-static "), as well as third-level and fourth-level categories based on the control method.

For the initial assault at Bourlon three divisions of Gruppe Arras under Otto von Moser were assigned.

For horticultural purposes, all Narcissus cultivars are split into 13 divisions by the Royal Horticultural Society, based partly upon flower form and partly upon genetic background.

For example, in Galpin's 1937 book A Textbook of European Musical Instruments, he lists electrophones with three second-level divisions for sound generation (" by oscillation ," " electro-magnetic ," and " electro-static "), as well as third-level and fourth-level categories based on the control method.

For most nations, cavalry was deployed in smaller units and was not therefore organized into divisions, but for larger militaries, such as that of the British army, United States, First French Empire, France, German Empire, Nazi Germany, Russian Empire, Empire of Japan, Second Polish Republic and Soviet Union, a number of cavalry divisions were formed.

For a significant part of the Christian world, one of the highest aims is the reconciliation of the various denominations by overcoming the historical divisions within Christianity.

For example, the Central Valley is a distinct region in itself both culturally and topographically from coastal California, though in Northern versus Southern California divisions, the Sacramento Valley and most of the San Joaquin Valley are usually placed in Northern California.

For elections to Nottinghamshire County Council the town is covered by two electoral divisions: Beeston North ( consisting of the North and West wards ) and Beeston South & Attenborough ( consisting of the Central, Rylands and Attenborough wards ).

For more information about the divisions of Polish lands in particular periods, see Administrative divisions of Poland (" Historical ").