SURETYSHIP. (15 SURFACE. ly against the part_v executing it. This will always be enforced when it appears that the lan- guage was chosen by the signer, whether he enters into the contract for his own benefit or for the benefit of a third person. Another important rule is that such contracts shall be interpreted so as to give effect to all of their provisions if possible. After the contract is made it is the duty of the creditor not to enter into any binding engage- ment with the principal, modifying that contract, without the assent of the surety. The law favors the surety and protects him with much jealousy. Accordingly, if the creditor varies the terms of the original contract or changes securities put into his hands by the principal debtor, or discharges a co-surety, or gives time to the principal debtor, or negligently causes a loss to the surety, the latter will be discharged unless he has assented to this conduct of the creditor. If, however, the principal debtor or a co-surety is discharged from liability by opera- tion of law, as by a discharge in bankruptcy, the suretj' still remains bound. The rights of the surety may be considered under three heads. First: Against the principal debtor. As soon as the debt becomes due, the surety is entitled to call on him for exoneration. This relief is obtainable in a court having equit- able powers, it being unreasonable that the .surety should have such a cloud hanging over him. If the surety has been compelled by the cred- itor to pay the debt he is entitled to call on the principal for reimbursement; for the money paid by him was paid for the principal's use. Second: Against the creditor. As soon as the debt be- comes due the surety may compel the creditor to sue the principal and collect the debt from him. In some of our States the surety is dis- charged from lialiility if the creditor does not sue the principal upon the suret_y's request. One who is surety for the honesty or good conduct of an employee is entitled to have the emplovee dis- charged from service for serious defaults or breaches of duty, or to be freed from his surety- ship. Another and ver.v important right of the surety is to have the benefit of all securities which the creditor holds against the principal. This is known as the right of subrogation (q.v. ). Third: Against co-sureties. It often happens that one surety is compelled by the creditor to pay the whole debt, and that the debtor is worthless. In such a case the unlucky surety is entitled to call upon his co-sureties for contribu- tion. See SuBROG.Tio:«. Consult authorities cited under GuAR.NTy. SURFACE (OF., Fr. surface, from Lat. superficies, upper side, surface, from super, above -{- fades, form, figure, face). Tlic boundary between two portions of space. As a point in a plane is determined in general by two inter- secting lines, so a point in space is in general determined by three intersecting surfaces. These surfaces niav be plane, quadric, or of higher order according as their equations are of the first, second, or higher degree in the linear coordinates of the system. Thus in Cartesian coordinates (see Coordinates) the general equa- tion of the first degree in x. y, s, or ax -- h;i + cz + d = 0. is represented by a plane. The gen- eral equation of the second degree in x. y. z. or ax- + hif -f C2= + Ifijz + 2p~x -f 27i.t;/ + 2k3S -- 2m!/ + 2nz + (? =; 0, is represented liy a coni- VoL. XVIII— 46. coid, or surface of the second order, also called a quadric surface. H,v a suitable transformation of coordinates the general e((uation of tlu' second degree may be transformed into one or the other of the forms (1) Ax' + iiir + CV = D or (2) Ajr + Bj/" = C'c. Surfaces having the symmetric equation (1) are synnnetric with respect to the origin as a centre and are called central quad- rics. Non-central quadrics are included in equa- tion (2). If A = B = 0, equation (1) takes the form a?' + j/^ -+- jr = K ( = r) , the equation of the sphere (q.v,). The general equation (1) represents either an ellipsoid (q.v.) or an hyper- boloid. If D = 0, and A, B, C are not all posi- tive, equation (1) represents a conical surface whose vertex is at the origin. Equation (2) is represented by the surface of a paraboloid (q.v.). A surface through everj' point of which a straight line may be drawn so as to lie entirely in the surface is called a ruled surface. Any one of these lines which lies on the surface is called a generating line of the surface. The , cylinder, cone, hyperboloid of one sheet, conoid (q.v.), and the hyperbolic paraboloid (see Para- boloid) are ruled surfaces. There are two dis- tinct classes of ruled surfaces, those on which the consecutive generators intersect and those on which they do not. The former are called developable and the latter skew surfaces. If the degree of the equation f(x,y,z) =: is higher than the second, the surface representing it will be of an order higher than the second. In discussing the properties of such surfaces, especially the nature of the surface in the vicin- ity of any given point, the equation of the tan- gent plane at that point is necessary. This plane is the locus of all tangent lines through the given point, and will meet the surface of the Jith order in a curve of the nth degree, since each straight line meets this curve in n points. The point of contact of the plane with the surface will be a singular point on the curve. (See Curve.) The section of any surface by a plane parallel and indefinitely near to the tangent plane at any point is a conic and is called the indicatrix at the point. Thus points of a surface are called ellip- tic, parabolic, or hyperbolic, according as the indicatrix is an ellipse, parabola, or hyperbola. If every straight line through a point (x', y', z') of a surface meets the surface in two coincident points, the point (x', y', z') is called a siiifjular point. If the tangent lines at any point form a cone the point is called a conical point: if they form two planes the point is called a nodal point. Similar to the envelope of a. family of curves, the envelope of a family of surfaces is the locus of the ultimate intersections of a series of sur- faces produced by varying one or more param- eters (q.v.) of an equation. The curve in which any surface is met by the consecutive surface is called the characteristic of the envelope. Every characteristic will meet the next in one or more points, and the locus of thesa is called the edge of regression or cuspidal edge of the envelope. The conditions for convexity and concavity, dif- ferent orders of contact, and various other prop- erties are best obtained from works on analytic geometry. Consult: Monge, Application de I'anah/se a la geometric (Paris, 1795) ; Dupin. Dtrcloppetnents de gfomftrie (ib., 1813) : Pliicker, 'Neue Ocome- trie des Raumes gegriindet auf die Betraehlung der geraden TAnie als Raumelement (Leipzig,
