Page:The New International Encyclopædia 1st ed. v. 05.djvu/145

* COEFFICIENT. 118 CCELENTEKATA. fore any letter. The coetlicients iii any alge- braic equation of the form j'fl + a,an -1 -f u^o -- + an =0 are related to the roots; thus, i + r-i + ■ ■ ■ r„=Si = Oi, An + >>'i + ■ ■■ I'n - 1 'a = S ii:^ = — a-,. ■ . /•[f, ... fn = ± <(d See Equation; for differential coeflicients, see Calculus; and for binomial coefficients, Bi- NOMLL TlIEORKM. Detached Coefficients. In many operations with algebraic functions, in which the letters are involved in ascending or descending powers, the required calculation may be performed with co- efficients only — e.g. to multiply x" + 3x-]/ + 3x1/- + i'^ by x" + 2x1/ + if 1 1 4- 3 + 3 + 1 2 ■ 2 + 6 -f (i + 2 1 1 - 3 4- 3 + 1 +5-1-10 + 10 + 5 + 1 Hence t' + ox'y + lOxY + lOj')/' + 5xy* + y' is the product. Indeterminate Coefficients. Jlany func- tions may be expanded to any desired number of terms by assuming arbitrary coefficients whose values are determined from resulting equations — e.g. to expand to 4 terms, assume ^ ^ ' 1— j;, = o + ij + car + dx" + c . . . 1 — J- Then, 1 = t( + ( h—a ) X + (c—h ) X- + (d—c) a;' + . . . If a = 1 , then 6 — (, = 0, c — 6 = 0, d — c = Q That is to say, if a = 1, 6 = a = 1, c = 6 — 1. (/ = c = 1. etc. Substituting these values in our assumed equation, we find; J—=: + x+x'+x^ + .... 1 — X The method is old as literal algel)ra. In physics, a constant expressing the measure of some property of a substance is often called a coefficient — e.g. the coefficient of elasticity is the quotient of the stress by the strain, or the quotient of the applied pressure by the voluminal compression produced. The coefficient of simple rigidity is the ratio of the shearing stress to the shearing strain. The coefficient of refraction (often called the index of refraction) is equal to the ratio of the speed of light in the first medium to its speed in the second. This ratio for air and water has the value 1..3.36, repre- senting the refractive power of water. The co- efficient of friction is the quotient of the resist- ance due to the sliding of one sulistance on an- other and the pressure producing the contact. The coefficient of expansion is the amount of expansion of a body of unit magnitude due to an increase of 1° in temperature. COEFFICIENT OF EXPANSION. See Heat. COEFFICIENT OF EXPANSION TOR GASES, Sec (Iasew, General Properties of. COEFFICIENT OF FRICTION, See Fric- tion, I COEFFICIENT OF SELF-INDUCTION, and Coefficient of Mutual Induction. See Electricity, and Induction. COEHOORN, ko'horn. A small mortar, usu- ally made of bronze, and formerly used in boats and small vessels and on the gangvaj'3 of larger .ships. It threw a shell of 12 to 14 pounds weight. Its name is derived from that of its in- ventor, Coehoorn (q.v,). I'or illustration, see Artillery, COEHOORN. kOo'hOrn, COEHORN, or CO'- HORN, .Mexno van (l(i41-1704). . Dutch military engineer. He was Iroru near Leeuwar- dcn, and aSter studying at the Cniversity of Franeker, entered the Dutch service at the early age of sixteen, as captain in an infantry regiment. He distinguished himself at Maes- triclit, Graave, and Senetle, in lt>7.'{-74. It was at the siege of Graave that he used with effect a mortar of his own design, later called after him. After the peace of Ximeguen (1G79) he was employed to fortify the various Dutch strong- holds. In I6S5 he published his important work on fortification, Xieuwe veslinyhuiiu:, which was speedily translated into French and German, Coelioorn was actively employed during the wars of William of Orange against French aggression, and at Kaiserswerth, Bonn, Fleurus, and Xamur lie showed himself no mean ojiponent of the great N'aul^an. He was made a lieutenant-general in 1095, and after the Peace of Ryswick (1697) was employed to refortify several of the Dutch towns. When the War of the Spanish Succes- sion began, Coehoorn was at once dispatched to the front, and, by his brilliant and energetic generalship, reduced six of the chief French strongholds. In 1704, while hastening to confer with ilarlborough at The Hague, he was stricken down ly apoplexy, ilarch 17th, Consult: De Peyster, Life of Coehoorn (New York, 1860); Bo- nomer, Essui general de fortification (Paris, 1814). CCELEBS (se'lebz) IN SEARCH OF A WIFE, A novel by Hannah Jloore (1809), giv- ing a bachelor's idea of a model wife. CCELENTERATA, se-len'te-ra'ta (Neo-Lat. nom. pi., from cwlenteron, from (Jk. Koios, hoilos, hollow + Hvrepov, enteron. intestine). One of the phyla of the animal kingdom, usually ranked as next to the lowest of the ty])es of Jletazoa; the polyps and jelly-fishes. It is characterized by the absence of a body-cavity and a separate cir- culatory system, both of these being functionally replaced by the system of chambers or tubes into which the mouth opens. Owing to the fact that this system is not only digestive, but that it takes the place of the bodj'-eavity, it is some- times called the coelenleron. ilore commonly the coelenteron is called the gastro-vascular cavity, since it not only serves as a stomach for diges- tion, but, by means of its branches, as a vascu- lar or circulatory system in conveying food (and perhaps oxygen) to all parts of the body. All of the Ctelenterata are more or less perfectly radially symmetrical, and generally on the number 4 or 6; that is, they can be divided in either four or six vertical planes, and the result- ing halves will be approximately similai". The Civlenterata may therefore be defined as radial- ly symmetrical, iinsegmented Metazoa, having a gastro-vascular cavity. In most Coelenterata there is no anus, waste matter from the digestive system being ejected through the mouth. Be- tween the endodermal lining of the gastro-vascu- lar canals and the ectodermal covering of the body is a structureless layer, known as the supporting layer or 'mesoglo?a.' This may be very thin and firm, or it mav be very thick and gelatinous. It