62 = φιν = φιφρ, for the vector of the second centre. Pages. (K2') (s). These expressions for σ1, 02 include (p. 689) a theorem of Dr. (t). In connexion with the same expressions for σ1, 2, it may be and r1 =fr=fUdp = (a2 — a22)-1 = &c., (L2) (L2') and σ2 with α1, this association of r1 and σ with a2, &c., and of r2 (u). By the properties of such surfaces, the scalar here called r2 is df Uvdp = 0, (M2) (v). Writing simply r and ✈ for r1 and r1⁄2, so that r' is constant, but variable, for a first line of curvature, while conversely r is con- stant and variable for a second line, it is found (pp. 684, 685, 586), that the scalar equation of the surface of centres (i) may be regarded as the result of the elimination of -1 between the two equations, 1=S.o (1 + r-1p)-2pσ, (N2), and 0=S.σ (1+r1p)-3p2σ; (N2′) whereof the latter is the derivative of the former, with respect to the scalar 1. It follows (comp. p. 688), that the First Sheet of the Sur- face of Centres is touched by an Auxiliary Quadric (N2), along a Quartic Curve (N2) (N2'), which curve is the Locus of the Centres of First Cur- vature, for all the points of a Line of Second Curvature; the same sheet being also touched (see again p. 688), along the same curve, by the developable normal surface (2), which rests on the same second line : with permission to interchange the words, first and second, through- out the whole of this enunciation. (w). The given surface being still a central quadric (0), the vec- tors p, σ, v can be expressed as functions of v (comp. (j) (k) (1)), * Dr. Salmon's result, that this surface of centres is of the twelfth degree, may CONTENTS. and conversely the latter can be expressed as a function of any one of the former; we have, for example, the reciprocal equations (p. 685), = (1 + r1p)-2 po; (02) o = (1 + r−16)2 4-1v, (02), and v= from which last the formula (N2) may be obtained anew, by observing (4) that Sov = 1. Hence also, by (r), we can infer the expres xxxix Pages. sions, p = (p −1 + r1) v = 2 ̄1 v, (P2), and v = -1 = P2 p = v2 ; (P2') and in fact it is easy to see otherwise (comp. p. 645), that v2 || 7 || v, and Spr2 = 1 Spv, whence v2 v as before. = (z). More fully, the two sheets of the reciprocal (j) of the surface of centres may have their separate vector equations written thus, and the scalar equation† of this reciprocal surface itself, considered as including both sheets, may (by page 685) be thus written, the functions ƒ and F being related as in 408, (b), with several equivalent forms; one way of obtaining this equation being the elimination of between the two following (same p. 685): Fv+r1 v2 = 1, (Q2); fv+rv2 = 0. (Q2') (y). The two last equations may also be written thus, for the first sheet of the reciprocal surface, F2 vi=1, (R2), and ƒʊv1 =r, in which (comp. pp. 685, 689), F2v=Svp21v= Sv (p−1 +r1) v ; (R2) (R2") and accordingly (comp. pp. 483, 645), we have F2v2 Fv= 1, and ƒʊn=fr=r. (2). For a line of second curvature on the given surface, the scalar r is constant, as before; and then the two equations (Q2), (Q2"), or (R2), (R2), represent jointly (comp. the slightly different enunciation in p. 688) a certain quartic curve, in which the quadric reciprocal (R2), of the second confocal (a2 b2 c2), intersects the first sheet (y) of the Reciprocal Surface (Q2); this quartic curve, being at the same time the intersection of the quadric surface (Q2′) or (R2), with the quadric cone (Q2′′) or (R2′), which is biconcyclic with the given quadric, fp=1. * The equation v = v2, = the normal to the confocal (a2 b2 c2) at P, is not actually given in the text of Series 412; but it is easily deduced, as above, from the formulæ and methods of that Series. + The equation (Q2) is one of the fourth degree; and, when expanded by coordinates, it agrees perfectly with that which was first assigned by Dr. Booth (see a Note to p. 685), for the Tangential Equation of the Surface of Centres of a quadric, or for the Cartesian equation of the Reciprocal Surface. ARTICLE 413.-On the Measure of Curvature of a Surface, The object of this short Series 413 is the deduction by quaternions, somewhat more briefly and perhaps more clearly than in the Lectures, of the principal results of Gauss (comp. Note to p. 690), respecting the Measure of Curvature of a Surface, and questions therewith con- (a). Let P, P1, P2 be any three near points on a given but arbitrary surface, and R, R1, R2 the three corresponding points (near to each other) on the unit sphere, which are determined by the parallelism of the radii OR, OR, OR2 to the normals PN, PIN1, P2 N2; then the areas of the two small triangles thus formed will bear to each other the ultimate ratio whence, with Gauss's definition of the measure of curvature, as the ultimate ratio of corresponding areas on surface and sphere, we have, by (b). If the vector p of the surface be considered as a function of two scalar variables, t and u, and if derivations with respect to these be denoted by upper and lower accents, this general transformation with a verification for the notation pqrst of Monge. (c). The square of a linear element ds, of the given but arbitrary and with the recent use (b) of accents, the measure (T2) is proved e,f,g; é,f,g'; fog.; and e,,-2f+g"; (U2') the form of this function (p. 692) agreeing, in all its details, with the (d). Hence follow at once (p. 692) two of the most important References are given, in Notes to pp. 690, &c. of the present Series 413, Is the Measure of Curvature at any Point, and IInd, the Total By a suitable choice of t and u, as certain geodetic co-ordinates, where t is the length of a geodetic arc AP, from a fixed point a to a variable point P of the surface, and u is the angle BAP which this variable arc makes with a fixed geodetic AB: so that in the immediate neighbourhood of A, we have n=t, and n' = Dm = 1. (). The general expression (c) for the measure of curvature takes thas the very simple form (p. 692), xli = Eis area being bounded by two geodetics, AP and AQ, which make with each other an angle Au, and by an arc ro of an arbitrary curve on the given surface, for which t, and therefore n', may be conceived to (9). If this arc pa be itself a geodetic, and if we denote by v the variable angle which it makes at P with AP prolonged, so that tan v Total Curvature of a Geodetic Triangle ABC=A+B+C−π, (V1⁄2′′) what may be called the Spheroidal Excess of that triangle, the total ARTICLE 414.-On Curvatures of Sections (Normal and Oblique) (a). The curvatures considered in the two preceding Series hav- ing been those of the principal normal sections of a surface, the present Series 414 treats briefly the more general case, where the section is made by an arbitrary plane, such as the osculating plane at p to an arbitrary curve upon the surface. (b). The vector of curvature (389) of any such curve or section being (-)-1=D,3p, its normal and tangential components are found - (W2') the former component being the Vector of Normal Curvature of the 694-698 g Surface, for the direction of the tangent to the curve: and the latter being the Vector of Geodetic Curvature of the same Curve (or section). points s and x, in which the axis of the osculating circle to the curve intersects respectively the normal and the tangent plane to the sur- face (p. 694); s is also the centre of the sphere, which osculates to the surface in the direction dp of the tangent; 01, 02 are the vectors of the two centres S1, S2, of curvature of the surface, considered in Se- ries 412, which are at the same time the centres of the two osculating spheres, of which the curvatures are (algebraically) the greatest and least and v is the angle at which the curve here considered crosses (d). The equation (W2) contains a theorem of Euler, under the it contains also Meusnier's theorem (same page), under the form (e). The expression (W2'), for the vector of geodetic curvature, ad- mits (p. 697) of various transformations, with corresponding expres- sions for the radius T(p − ) of geodetic curvature, which is also the radius of plane curvature of the developed curve, when the developable circumscribed to the given surface along the given curve is unfolded into a plane and when this radius is constant, so that the developed curve is a circle, or part of one, it is proposed (p. 698) to call the given curve a Didonia (as in the Lectures), from its possession of a certain iso- perimetrical property, which was first considered by M. Delaunay, or 8fS (Uv.dpop)+co (Tdp = 0; (X2) (X2) by the rules of what may be called the Calculus of Variations in Qua- ternions e being a constant, which represents generally (p. 698) the radius of the developed circle, and becomes infinite for geodetic lines, which are thus included as a case of Didonias. ARTICLE 415.-Supplementary Remarks, (a). Simplified proof (referred to in a Note to p. xii), of the gene- ral existence of a system of three real and rectangular directions, which satisfy the vector equation Vpop = 0, (P), when o is a linear, vector, and self-conjugate function; and of a system of three real roots of the cubic equation M=0 (p. xii), under the same condition (pp. 698- |