Concerning this integration, the author obferves that, unlefs the exponent p be lefs than n, the given fraction will be an improper one; in which cafe the whole quantities, contained in it, must be feparately integrated, and added to the integral afterward found. Another condition is that p be a whole number: but fhould it be a fraction, M. EULER points out a method of transforming this expreffion into another, in which the exponents are whole numbers. This mode is also applicable in thofe cafes in which p has an imaginary value, provided it be fuch that the differential formula be real: the only cafe in which it cannot be used, is when p-n is a real and affirmative quantity.

Concerning the Neceffity of having recourse to imaginary Quantities in the Integration of differential Formula. By the fame.

This ingenious mathematician had endeavoured to find a method of integrating rational formula without making use of imaginary quantities: in fome cafes he fucceeded, but failed in others, particularly in that in which the denominator has two or more factors. A cafe is here investigated, in the integration of which, imaginary quantities feem indifpenfably neceflary.

A Problem in the indeterminate Analysis of Infinites. By the fame.

This problem is to find fuch a relation between the variable quantities and z that the expreffion qz may be algebraical, and that the formulæ fz(99-1) fhall exprefs an

are of a circle. As the latter condition is the most difficult, the

author begins with it, by making (az√(99—1) =A. tang.

X

y

whence, by taking the differentials, and affuming xx+yy=zz, удх-хду Z√/(1+PP) he deduces ✔(99—1)=x5x+ysy and q=

бу

ΟΙ

ay. Thus the two variable quantities, q and z, are de

P = dx

termined by means of x and y, and the second condition is fulfilled. In order to comply with the first, as qdz=dx√✓(1+pp) and dypox, he proceeds to render these two expreffions algebraical, making y=fpx, and fqdz=Sdx/(1+pp). Hence Spdx-px-Sx&p;

xpdp

and fox/(1+pp)=x√/(1+pp)—s ✓ (1+pp)

of thefe formulæ two folutions are here given, the one by furds, the other by rational quantities.

This is indeed an indirect folution of the problem but it is published only with an intention to affift mathematicians in their farther endeavours toward the improvement of this branch of analyfis.

Concerning Rectifiable Lines geometrically drawn on the Surface of a Spheroid. By the fame.

As we hitherto know of no rectifiable line that can be drawn on the furface of the sphere, except an epicycloid generated by the motion of a great on a small circle, the radii of which are in a rational ratio to each other, it might be thought still more difficult, on account of the elliptic arcs, to discover a rectifiable line on the furface of a fpheroid: but a theorem is here given, by which its ingenious author not only facilitated the folution of this problem relative to the fphere, but was also enabled to extend its application to the fpheroid.

He obferves, that if v denote any function of the angle O, which the normal of a curve makes with the axis of its abfciffe,

the element of the arc being ds=v dø +d!

the curve may be thus expreffed:

the ordinates of

If therefore be an algebraical function, not of the angle, but of its tangent t, the ordinates x and y will be expreffed by algebraical functions of t, and the curve will be algebraical and rectifiable. By means of this theorem, an infinite number of algebraical curves may be determined, which will not only be rectifiable, but the rectification of which will depend on a given quadrature.

2

After demonftrating this theorem, the author proceeds to the investigation of rectifiable lines on the furface of a spheroid, the axis of which is to its equatorial diameter, as c: 1; and, as the element of a line drawn on fuch a fpheroid is ds=✓✓/dx2+dy+dz, x, y, and z, being the three ordinates, it is evident that nothing more is requifite than to find such a relation between x and y, as fhall render this expreffion integrable-but as this cannot be done univerfally, the reader's attention is directed to a particular cafe, in which s=nz; and thus rectifiable lines are found, which may be drawn on the furface of the fpheroid, and which have this property, that their projection on the plane of the equator is the fame, whether the spheroid be oblate or prolate; and, as the quan

L13

tity

tity c, which determines the femi-axis of the generating ellipfis, is thrown out in the calculus, this folution is also applicable to the hyperbolic conoid.

Concerning the Difficulties which occur in investigating the Superfices of the Scalene Cone. By the fame.

As this memoir contains rather an enumeration, than a solution, of thefe difficulties, we fhall not detain our readers with an attempt to analyse it.

Concerning fome Properties of Ellipfes defcribed on the Surface of the Sphere. By M. N. Fuss.

It is here supposed that a thread, the length of which is 2c, is fixed in two points of the fuperficies of the sphere, the distance of which from each other, is an arc of a great circle 2a, and that, by extending this thread with a drawing pen, an ellipfis is defcribed: then, if the abfciffe be affumed on the great circle in which the foci are taken, and computed from the half of the diftance of these from each other, calling the abfciffa x, and the ordinate y, the following equation will be found:

tang. y=√(fin. c2—fin. a2) (fin. c2—fin. x2

Sin. c. cof. c.

This expreffion, when applied to a particular cafe, in which the length of the thread is equal to half the circumference of the fphere, leads to the discovery of a remarkable property;` which is that, in this cafe, whatever may be the distance between the two points affumed, the curve defcribed will always be a great circle of the fphere.

In general, if half the conjugate diameter of an ellipfis on the sphere be called g, its equation will be

tang. g. tang. y fin. c.

-√(fin. c2—fin. x2).

In order more accurately to inveftigate the nature of this curve, M. Fuss examines its projection on the plane of a great circle, the pole of which is the centre of the ellipfis. This projection is an ellipfis; and if we put G for half the conjugate, and C for half the transverse diameter, calling the abfciffa from the centre X, and the ordinate Y, its equation will be G Y=√(CC-XX). The connection of this equation with that of the ellipfis on the sphere is fufficiently obvious.

Obfervations on the fixteenth Theorem of the first Book of Pappus Alexandrinus. By M. F. I. SCHUBERT.

This memoir contains an analytical investigation of this propofition; that, of all circular areas of the fame circumference, that of the femicircle is the greatest.

PHYSICO

PHYSICO-MATHEMATICAL CLASS.

Concerning the Centripetal Forces, requifite to aefcribe Curves, which are not in the fame Plane. By M. EULER.

In the problem, folved by Sir Ifaac Newton, requiring the centripetal force neceffary to make a body defcribe a given curve, this is fuppofed to be in the fame plane with the centre of force fought :-but it is evident that, if the curve be not in the same plane, the motion cannot be effected by a fingle centripetal power, and requires at least two centres of force.

It is here fhewn, that, whatever be the curve described, these two centres may be affumed at pleasure; and that, from thefe, the two centripetal forces, necessary for the description of the curve, may be determined.

By decompofing the motion of the body into three directions, parallel to three axes affumed at pleasure for the co-ordinates of the curve, three equations may be found for the three accelerating forces; and hence may be deduced the two centripetal forces required, together with the velocity of the body in every point of the curve. This folution becomes ftill more fimple, if the axis of the abfciffæ be made to pass through the centres of force, and the commencement of the abfciflæ be taken at half the distance between them. This is illuftrated by two examples. In one of thefe, the centres of force are affumed in the axis of a cylinder, on the furface of which the body is fupposed to move; in the other, the body is confidered as moving on a sphere, with the centres of force in its poles; and here that cafe is particularly investigated, in' which the curve described is a loxodromic.

Concerning the Motion of three Bodies mutually attracting each other in a right Line. By the fame.

We have here an ingenious attempt to folve this problem analytically, which cannot be done unlefs the two intermediate distances remain in a conftant given ratio to each other.

Solution of a Problem in Mechanics. By the fame.

M. EULER here fuppofes two cylinders, lying parallel to each other on a horizontal plane, with a thread wound around them, which paffes over the one, and under the other, in a contrary direction: each cylinder is reprefented by the circular plane of its fection, and thefe, together with the thread, are in the fame vertical plane. He then inveftigates the laws, according to which thefe cylinders will move, if an impulse be given, either to one of them, or to both at the fame time. In doing this, he difplays great ingenuity, and has reduced the whole queftion to the integration of an eafy differential equation.

On the gyratory Motion of a Body fastened to an extenfible Thread. By M. JAMES BERNOULLI.

We endeavoured to give the reader fome idea of M. BERNOULLI's former refearches on this fubject: but the present memoir affords fo little fatisfaction in the refult, that we shall not involve ourselves in the inextricable labyrinth of differential equations, to which it leads. The thread, to which the body is faftened, is here fuppofed to make an acute angle with a vertical line, around which the body has a turbinatory motion, and, by the alternate extenfions and contractions of the threads, defcribes a line of double curvature: but, after pursuing this queftion in a manner that difplays great fagacity and perfeverance, M. BERNOULLI ingenuously acknowleges that the equations, to which it leads him, are fuch as incline him to fufpect fome error, either in the hypothefis, or in the calculation, and that he publishes the memoir folely with a view to recommend the inquiry to other mathematicians.

Concerning the Dioptric Elements of the Theory of Achromatic Object-glaffes for Microfcopes. By M. W. L. KRAFFT.

The principles which M. Euler laid down for the theory of achromatic telescopes are here applied to microscopes. The general elements, on which the theory is founded, exprefs the angle, under which the axis of the lens is interfected by the refracted ray, and the diftance of the point of interfection from the ultimate refracting furface. In order to render thefe formulæ applicable to every point of the object, M. KRAFFT represents the position of the luminous point by two co-ordinates, one of which determines its horizontal distance from the lens, and the other its vertical height above the axis. Having demonftrated thefe formulæ, he applies them to rays infinitely near the axis, to the aberration occafioned by the fphericity of the lens, and, lastly, to the coloured rays, and to the aberration refulting from their various degrees of refrangibility.

PHYSICS.

On the Mufcular Fibres of the Heart; fixth and seventh Differtation. By M. C. F. WOLFF.

This gentleman proceeds very flowly, and with the utmost minutenefs, in his anatomical defcription of the heart; and, if he continues to treat the fubject in the fame manner, it may furnish matter for many more memoirs. The former of thefe differtations is a continuation of that mentioned in our laft appendix, and defcribes the fibres of the left ventricle; the other contains a general account of the feveral ftrata of fibres. M. WOLFF obferves, that none of these ftrata, except the exterior and interior, extend all over the heart; the intermediate,

though