Ernest William Hobson

“ He used the notation sin, tan, sec for the sine, tangent, and secant of an arc. In the second half of the 17th century the theory of infinite series was developed by Wallis, Gregory, Mercator, and afterwards by Newton and Leibnitz. In the Analysis per æquationes numero terminorum infinitas, which was written before 1669, Newton gave the series for the arc in powers of its sine; from this he obtained the series for the sine and cosine in powers of the arc; but these series were given in such a form that the law of the formation of the coefficients was hidden. ”

“ In the wider sense of the term, a spherical harmonic is any homogeneous function of the variables which satisfies Laplace's equation, the degree of the function being not necessarily integral or real, and the functions are not necessarily rational in x, y, z, or single-valued ”

“ The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science, which is from analogy frequently called spheroidal trigonometry. ”

“ Expansion of Functions in Series of Legendre's Coefficients.—If it be assumed that a function given arbitrarily in the interval to , can be represented by a series of Legendre's coefficients and it be assumed that the series converges in general uniformly within the interval, the coefficient a can be determined by using (19) and (20) ; we see that the theorem (19) plays the same part as the property , does in the theory of the expansion of functions in series of circular functions. ”

“ The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality. ”

“ Spherical and other harmonic functions are of additional importance in view of the fact that they are largely employed in the treatment of the partial differential equations of physics, other than Laplace's equation; as examples of this, we may refer to the equation , which is fundamental in the theory of conduction of heat and electricity, also to the equation , which occurs in the theory of the propagation of aerial and electro-magnetic waves. ”

“ The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved right-angled plane and spherical triangles by geometry.
The Arabs were acquainted with Ptolemy’s Almagest, and they probably learned from the Indians the use of the sine. ”

“ Heine showed (Crelle’s Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier’s Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (Crelle’s Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math. Ann. vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. ”

“ Euler's great improvement consisted in his regarding the sine, cosine, &c., as functions of the angle only, thereby giving to equations connecting these functions a purely analytical interpretation, instead of a geometrical one as heretofore. The exponential values of the sine and cosine, De Moivre's theorem, and a great number of other analytical properties of the trigonometrical functions are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy. ”

“ Every new class of surfaces which may be considered would have in this extended sense a trigonometry of its own, which would consist in an investigation of the nature and properties of the functions necessary for the measurement of the sides and angles of triangles bounded by geodesics drawn on such surfaces. ”

“ The ordinary spherical harmonics of positive integral degree n are those which are rational integral functions of x, y, z. The most general rational integral function of degree n in three letters contains coefficients; if the expression be substituted in (1) , we have on equating the coefficients separately to zero relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains , or independent constants. There exist, therefore, independent ordinary harmonics of degree n ”

“ The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. ”

“ For the treatment of potential problems which relate to spaces bounded by special kinds of surfaces, solutions of Laplace's equation are required which are adapted to the particular boundaries, and various classes of such solutions have thus been introduced into analysis. Such functions are usually of a more complicated structure than ordinary spherical harmonics, although they possess analogous properties. As examples we may cite Bessel's functions in connexion with circular cylinders, Lamé's functions in connexion with ellipsoids, and toroidal functions for anchor rings. ”

“ D’Alembert showed that the ordinate y at any time t of a vibrating cord satisfies a differential equation of the form δ2y/δt2 = a2δ2y/δx2, where x is measured along the undisturbed length of the cord, and that with the ends of the cord of length l fixed, the appropriate solution is y = ƒ (at + x) − ƒ (at − x) , where ƒ is a function such that ƒ (x) = ƒ (x + 2l) ; in another memoir in the same volume he seeks for functions which satisfy this condition. In the year 1748 (Berlin Memoirs, vol. iv.) Euler, in discussing the problem, gave ƒ (x) = α sin πx/l + β sin 2πx/l + . . . ”

“ The Indians, who were much more apt calculators than the Greeks, availed themselves of the Greek geometry which came from Alexandria, and made it the basis of trigonometrical calculations. The principal improvement which they introduced consists in the formation of tables of half-chords or sines instead of chords. ”

“ The question was then considered from a new point of view by Lagrange, in a memoir on the nature and propagation of sound (Miscellanea Taurensia, 1759; Œuvres, vol. i.) , who, while criticizing Euler’s method, considers a finite number of vibrating particles, and then makes the number of them infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli’s Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous ”

“ TRIGONOMETRY 571 connected with the circle. We may easily show from the definitions that cos 2 (a; + iy) + sin* (x + iy) = 1, cosh 2 y - sinh - y 1 ; cos (ar + iy) = cos x cosh y - 1 sin x sinh y, sin (x + iy) = sin x cosh y + 1 cos x sinh y, cosh (a + |3) = cosh a cosh /3 + sinh a sinh /3, sinh (a + /3) = sinh a cosh - cosh a sinh /3. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. Expan- If we equate the coefficients of 71 on both sides of equation (13) , sionofan we set . . ”

“ This remains true however many partial intervals are taken, and therefore, when their number is increased indefinitely, and their breadths are diminished indefinitely according to any law, we have
when M is intermediate between the greatest and least values which can have, when x is in the given integral. Now this integral is a continuous function of its upper limit x, and therefore there is a value of x in the interval, for which it takes any particular value between the greatest and least values that it has. ”

“ Their objection appears, however, to rest upon a misapprehension as to the meaning of the sum of the series; if x1 be the point of discontinuity, it is possible to make x approach x1, and n become indefinitely great, so that the sum of the series takes any assigned value in a certain interval, whereas we ought to make x = x1 first and afterwards n = ∞, and no other way of going to the double limit is really admissible. ”

“ The sin A sin B sin C equation a = b cos C+c cos 5 expresses the fact that the side a is equal to the sum of the projections of the sides b and c on itself ; thus we obtain the equations b=c cosA + acosC -. c=acos5+6cos^ J If we multiply the first of these equations by -a, the second by b, and the third by c, and add the resulting equations, we obtain the formula Z> 2 + c 2 -a 2 =2&ccos^ or cos A= - - , which gives the cosine of an angle in terms of the sides. ”

“ The operation
performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis , and is denoted by . The potential function is defined to be the potential due to a singular point of degree zero at the origin; is called the strength of the singular point. Let a singular point of degree zero, and strength , be on an axis , at a distance from the origin, and also suppose that the origin is a singular point of strength ; let be indefinitely increased, and indefinitely diminished, but so that the product is finite and equal to ”

“ Let H denote the radius of the circumscribed circle, r of the in- Radii of scribed, and r lt r 2 , r 3 of the escribed circles of a triangle ABC ; the circum- values of these radii are given by the following formulae. _ _ a be _ a ~ = n r=- = (s-a) tan A = s Of sin B sinj C, = - = s tan A = iB sinj A cosj B cos J C. s a scribed, in scribed, and escribed circles of a tri angle. Spherical Trigonometry. We shall throughout assume such elementary propositions in spherical geometry as are required for the purpose of the investiga tion of formulae given below. ”

“ Each time that the straight line makes a complete revolution round we consider it to have described four right angles, taken with the positive or negative sign according to the direction in which it has revolved ”

“ If we write , the integral becomes , which is the form in which the integral is frequently considered.
The Second Mean-Value Theorem.—The limiting value of Dirichlet’s integral may be conveniently investigated by means of a theorem in the integral calculus known as the second mean-value theorem. Let a, b be two fixed finite numbers such that a < b, and suppose ƒ (x) , φ (x) are two functions which have finite and determinate values everywhere in the interval except for a finite number of points ”

“ Ann., vol. vi., 1872) and by Hölder (Göttinger Anzeigen, 1894) may be consulted; the most general form of the theorem has been given by Hobson (Proc. London Math. Soc., Series II. vol. vii., 1909) . On the theory of uniform convergence of series, a memoir by W. F. Osgood (Amer. Journal of Math. xix.) may be with advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (Annali di matematica, Series III. vol. iii.) is of great importance. ”

“ The degree n may be fractional or imaginary, but we are at present mainly concerned with the case in which n is a positive or negative integer. If x, y, z be replaced by their values , , in polar co-ordinates, a solid spherical harmonic takes the form ”

“ If x, y, z are such as to satisfy the two equations , , the equation is also satisfied; the problem of determining the poles is therefore equivalent to the algebraical one of reducing to the product of linear factors by means of the relation , between the variables. Suppose
we see that the plane passes through two of the 2n generating lines of the imaginary cone in which that cone is intersected by the cone . ”

“ A memoir by Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue’s work on Fourier’s Series will be found in his treatise, Leçons sur les séries trigonométriques (1906) ; also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903) , and in a paper “Sur la convergence des séries de Fourier,” Math. ”

“ This function , thus defined for , is of considerable importance in the potential theory. When , we may in a similar manner obtain two series in ascending powers of , one of which represents , and a certain linear function of the two series represents the analytical continuation of , as defined above. The complete primitive of Legendre's equation is
By the usual rule for obtaining the complete primitive of an ordinary differential equation of the second order when a particular integral is known, it can be shown that (27) is satisfied by
the lower limit being arbitrary. ”

“ In case the orthogonal surfaces are concentric spheres, co-axial, circular cones, and planes through the axes of the cones, the parameters are the usual polar co-ordinates r, , , and in this case , thus Laplace's equation becomes
Assume that is a solution, R being a function of r only, of only, of only ”