Any line (8) upon this surface is considered to be a screw, of which the pitch is c+a cos 20, where c is any constant, and is the angle between 8 and the axis of x. The fundamental property of the cylindroid is thus stated. If any three screws of the surface be taken, and if a body be displaced by being screwed along each of these screws through a small angle proportional to the sine of the angle between the remaining screws, the body after the last displacement will occupy the same position that it did before the first. For the complete determination of the cylindroid and the pitch of all its screws, we must have the quantities a and c. These quantities, as well as the position of the cylindroid in space, are completely determined when two screws of the system are known. In the model of the cylindroid which was exhibited, the parameter a is 2.6 inches. The wires which correspond in the model with the generating lines of the surface represent the axes of the screws. The distribution of pitch upon the generating lines is shown by colouring a length of 2-6× sin 20 inches upon each wire. The distinction between positive and negative pitches is indicated by colouring the former red and the latter black. It is remarkable that the addition of any constant to all the pitches attributed in the model to the screws does not affect the fundamental property of the cylindroid. When a rigid body is found capable of being displaced about a pair of screws, it is necessarily capable of being displaced about every screw on the cylindroid determined by that pair. The theorem of the cylindroid includes, as particular cases, the well-known rules for the composition of two displacements parallel to given lines, or of two small rotations about intersecting axes. If the parameter a be zero, the cylindroid reduces to a plane, and the pitches of all the screws become equal. If the arbitrary constant which expresses the pitch be infinite, we have the theorem for displacements, and if the pitch be zero, we have the theorem for rotations. As far as the composition of two displacements is concerned, the plane can only be regarded as a degraded form of the cylindroid from which the most essential features have disappeared. On the Number of Covariants of a Binary Quantic. By Professor CAYLEY, D.C.L., F.R.S. The author remarked that it had been shown by Prof. Gordan that the number of the covariants of a binary quantic of any order was finite, and, in particular, that the numbers for the quintic and the sextic were 23 and 26 respectively. But the demonstration is a very complicated one, and it can scarcely be doubted that a more simple demonstration will be found. The question in its most simple form is as follows: viz. instead of the covariants we substitute their leading coefficients, each of which is a "seminvariant" satisfying a certain partial differential equation; say, the quantic is (a, b, c....kx, y)", then the differential equation is (adb+2bdc....+njdk)u=0, which quà equation with n+1 variables admits of n independent solutions: for instance, if n=3, the equation is (adb+2bde+3cdd)u=0, and the solutions are a, ac-b2, a'd-3abc+263; the general value of u is u= any function whatever of the last-mentioned three functions. We have to find the rational non-integral functions of these functions which are rational and integral functions of the coefficients; such a function is and the original three solutions, together with the last-mentioned function a22+&c., constitute the complete system of the seminvariants of the cubic function; viz. every other seminvariant is a rational and integral function of these. And so in the general case the problem is to complete the series of the n solutions a, ac-b2, a'd-8abc+2b3, a3e—4a2bd+6ab2c-3b, &c. by adding thereto the solutions which, being rational but non-integral functions of these, are rational and integral functions of the coefficients; and thus to arrive at a series of solutions such that every other solution is a rational and integral function of these. On a Canonical Form of Spherical Harmonics. By W. K. CLIFFORD, B.A. The canonical form in question is an expression of the general harmonic of order n as the sum of a certain number of sectorial harmonics, this number being, when 5п-10 5n-9 n is even, and when n is odd, 2 2 Laplace's operator, ddy+dz may be obtained from the tangential equation d d d of the imaginary circle 2+n+2=0, by substituting de du dz dx' dy for έ, n, . If, therefore, a form U=(x, y, z)" is reduced to zero by this operator, it follows from Prof. Sylvester's theory of contravariants that the curve U=0 is connected by certain invariant relations with the imaginary circle. I find that U can be expressed in the form U=A"+B"+C”+.. where A=0, B=0,.... are great circles touching the imaginary circle, the number of terms being as above. Now if L=0, M-0 be two such great circles meeting in a real point a, and if ø be a longitude and 6 latitude referred to a as pole, it is easy to see that L”+M"=/ sin” 0 sin no+m sin” e cos no, a sum of two sectorial harmonics, which is the proposed reduction. When n is less than five, exceptions of interest occur. For n=3, if we take a, b, corresponding points on the hessian of the nodal curve U=0 (Thomson and Tait, Treatise on Natural Philosophy, § 780), and if we call 1, 4, the longitudes, 0, 0, the latitudes referred to these poles, we have Usin3 0, sin 30, +m sin3 0, cos 30 +n sin3 0, sin 342 +8 sin3 6, cos 302 For n=4, the nodal curve is of the species first noticed by Clebsch, of which many most beautiful properties have been pointed out by Dr. Lüroth. The form U is expressible as the sum of five fourth powers; so that if we take a, b real points of intersection of two pairs of them, c a real point on the fifth, calling 1, P2 Pv 01, 02, 0, longitudes and latitudes referred to them, we have On certain Definite Integrals. By J. W. L. GLAISHER, B.A., F.R.A.S. The integrals So sin (r")dx, cos (xn)dx have been evaluated in several different ways, and the investigations all present points of interest. The integrals have XP-1 cos x dx, deducible by an obvious transformation; and so universally have the latter forms been adopted, that the former are not to be found in Prof. De Haan's Tables. usually been written in the forms Xp-1 The most natural way of obtaining sin anda and cos xndr is by writing p=i(= √−1) in the well-known form of the Gamma Function, and we can equate real and imaginary terms. A curious difficulty, however, here presents itself, viz. to decide what value k (which must be integral) has. In a similar case De Morgan determined the proper value of k in the following manner :Put p=cos i sin ĕ, then Now if 0=0, the last integral vanishes; so that we must have k=0, and therefore obtained by differentiating S The above investigation is, however, chiefly valuable as suggestive of the result; it contains no indication of the limits between which a must lie that the last written equations may be true and the integrals not infinite. The integrals have also been e-ax sin xdx with regard to a and putting a=0 afterwards; but the results obtained are of the form an sin adr (n integral), and must therefore be infinite. The following investigation of the values of the integrals seems of interest, as it is rigorous and discriminates between the finite and infinite values. Integrate "sin y da dy with regard to x first, and we find it S * This method is also given in De Morgan's 'Differential and Integral Calculus,' pp. 630, 576. Some analysts (Oettinger, Bidone, &c.) have not seen any objection to a sin ædæ being finite for all values of n; but unless we are prepared to write with De Morgan ("Theory of Probabilities," Encyc. Met. p. 436)| ePdx=see how this can be admitted. because e-pxdx = it is difficult to P =1 (in which 2n> 1, so that m may have any value except such n as lie between 1 and -1), and using the relation г(α)г(1—α)=π COSеC άя, we The author had calculated a Table of the values of sin anda, cos anda for different values of x; and the curves y=sin arda, y = cos aada, as obtained from them, were drawn and exhibited to the Section, the discontinuities in each being remarked. [The Tables and curves will be found in the 'Messenger of Mathematics,' 1871.] On Lambert's Proof of the Irrationality of, and on the Irrationality of certain other Quantities. By J. W. L. GLAISHER, B.A., F.R.A.Š. The arithmetical quadrature of the circle, that is to say, the expression of the ratio of the circumference to the diameter in the form of a vulgar fraction with both numerator and denominator finite quantities, was shown to be impossible by Lambert in the 'Berlin Memoirs' for 1761; and the proof has since been given in an abridged and modified form by Legendre in the Notes to his 'Éléments de Géométrie.' Although Legendre's method is quite as rigorous as that on which it is founded, still, on the whole, the demonstration of Lambert seems to afford a more striking and convincing proof of the truth of the proposition; his investigation, however, is given in such detail, and so many properties of continued fractions, now well known, are proved, that it is not very easy to follow his reasoning, which extends over more than thirty pages. The object of the present paper is to exhibit Lambert's demonstration of this important theorem concisely, and in a form free from unnecessary details, and to apply his method to deduce some results with regard to the irrationality of certain circular and other functions. The theorem which Lambert proves, and from which he deduces the irrationality of π, is that the tangent of a rational arc (i. e. an arc commensurable with the radius) must be irrational; and this he demonstrates by means of an expression for the tangent as a continued fraction, viz. adopting an established notation for continued fractions in which that which follows each minus-sign is written as a factor, to save room. Consider a continued fraction P Suppose also that the continued fraction (i) is equal to be such that Q' and let R1, R.... Rn... R1 =α1P-B1Q, Rq=a2R1+B2P, R3=3R2+B3R1, Rn=anRn−1+ßnRn−2, then R-P-P,Q, as can be shown by induction; so that If P and Q be integers and 1....... an....... .B1... ẞn... be also all integers, then from the equations by which R1... Rn. are determined, we see that they also are integers. ... Now in the case of the continued fraction for tan ar=(2n−1)y, q=(2n-1)yqn-1-x2-2; ... ... and we notice that if x and y be integers, then a ... an... B1. Bn.. are so too, and consequently (if P and Q are integers) R1... Rn... are integers. The factor by which B1...Br is multiplied to obtain B1... Br+1 is gr+1 which can be made as small as we please by increasing r. We can therefore from (v), Q being finite, make Rn as small as we please by taking n sufficiently large; but if P and Q be both integers, Rn must remain an Р integer whatever value n may have; thus if be rational, (= y rational; but tan=1, so that cannot be rational. 4 tan must be ir The above is in substance Lambert's demonstration; alterations have been made in points of detail &c., and the notation has been changed. It may be noticed that the proof does not (as of course it should not) hold good if P and Q be infinite integers; for we cannot make Rn as small as we please in (v) if Q be infinite. |