i \. o o-^ v ' ( )^'( y#n — cos vf/ sin ({>) + ?/ (cos 0 cos \p cos

ft c A V=Vo = COS n sin w C . , C — A > * + — x -y . „ , , c cos ui A n, c, y, &, \^ 0 , and

(z/ - z,) {(x, - X/) 2 + (y, - y!Y + {z t - zifY ( x i v! — Vi x !) j*l'3 f, . 3 (*,*/ + y,y/ + *,*/) , „ 1 1 + ~i + &c, •} d m By the properties of the principal axes J x j y l dm = d,J'x i z t dm = d>fy, z, dm = 0, and by the properties of the centre of gravity J' x l dm = 0, fy t d m = 0, fz,dm = 0, whence 22 MR. LUBBOCK’S RESEARCHES Cdr + (B — A)pqdt = 3 - M - ( '^ i — — x,y/At Bdq + {A- C)rpdt = 3 -—- C) z/x/d f ^4dp + (C- B)yrd< = ^-^ ^y/z/dt Substituting for xj, y\, zj their values from equations, p. 20, upon the sup- position of 4 1 = 0, Cdr + (B — A)pqdt — 3 M ^ ~ ^ j { (y 1 cos 0 — s' sin 0) 2 — z' 2 } sin 2 ?> + 2 x'( 2 /' cos 0 — s' sin 0) cos' 2

If Adp + (C — B)qrdt — 3M r ^ - | (t/' cos 0 — z' sin 0) (^' sin 0 + z' cos 0) | = | x ' s ‘ n ® + * f cos A) ^ = -P' Bd g + (.4 — C) rpd* = (^4 — C) d < {P'cos p + Psin A dp + (C — B) qr dt = (C — B) dt {P cos

+ it + e) + (k — k') sin (

dy izdx + ^7 dy + Y~ z dz or, which is the same thing, XrfcT + Yrfy-f- Zrf* = 0 ; and if ds represent the distance of the two points, we obtain x£+y£+z£=o. ds d s dx dy dz ds ’ ds ’ ds Now 3-r, are the cosines of the angles which the directions of the forces make with the line d s ; wherefore the expression on the left side of the foregoing formula is the sum of the partial forces which act in the direction of ds ; and as this sum is equal to zero in all positions of the line ds round the point (a, y, z), the resultant of the forces produces no effect in the plane touch- ing the surface, and consequently its whole action is perpendicular to that plane. The nature of the case requires further, that the same resultant be directed towards the surface of the fluid. What has been deduced from the algebraic expressions is evident in another view. For, could we suppose that the resultant of the forces is not at every point perpendicular to the surface at liberty, it might be resolved into two partial forces, one acting in the tangent plane, and the other perpendicular to that plane ; and as the first force is opposed by no obstacle, it would cause the particles to move, which is contrary to the equilibrium. If we suppose that p is constant in the general formula of the hydrostatic pressure, we shall have an equation,

-|-f(XS.r-l-Y&3/-|-Z&2) = 0. ( 3 ) AND THE FIGURE OF A HOMOGENEOUS PLANET. 115 This equation must hold at every point of the mass of fluid without any rela- tion being supposed between the variations, wherefore p must be a function of three independent variables ; and in consequence the foregoing equation im- plies the three separate equations following, viz. d JP__ v _ dx @ J 5 dy fY, It now appears that the conditions of integrability must be fulfilled, viz. rf.gX d.g Y d.gX. d.g Z d.g Y d.gTj dy dx 5 dz dx ’ dz dy and unless the forces possess the properties expressed by these equations, the equilibrium will be impossible. Without pursuing the investigation in all its generality, we shall confine our attention to the case in which 'K.dx + Y dy Zi dz, is an exact differential ; a supposition that comprehends all the applications of the theory. If we represent the integral of the differential by p X u -f- f dm = 0 ; consequently, * , fdm S?+ =°; and by taking the sum of the similar quantities in all the parts of the canal, we obtain / Sp+ //^=°. Butp being a function of three independent variables, the sum of its variations, supposing the flowing quantities to follow any arbitrary law of increase or de- crease, is equal to the difference of p' and p°, the final and initial values of the function ; wherefore we have p' -p° +f-~r = <>■ Now /dm, that is the quantity of matter multiplied by the accelerating force, is the impulse or pressure in the direction of the canal caused by all the forces urging d m ; and as this pressure is exerted on the surface u, is the same pressure reduced to the unit of surface. Therefore, whatever be the figure of the canal, it follows from the foregoing investigation, that the difference of the pressures at its two extremities is equal to the sum of the impulses of all the contained molecules of fluid, every impulse being reduced to the direction of the canal and to the unit of surface. If the extremities of the canal be both in the parts of the outer surface which are at liberty, the pressures p' and p° will be both evanescent, and there will be no effort of the fluid either way, and no tendency to run out at one end. Further, if a canal be continued through the fluid till it return into itself, the 118 MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, initial and final pressures being the same, the impulses of the molecules in the whole circuit will balance one another. But in this case, the reasoning we have employed will not be exact, unless p, the algebraic expression of the pressure, be such a function as admits of only one value for any three given co-ordinates ; a restriction however, which, in every point of view, seems in- dispensable. 6. The whole theory, it will readily appear from the foregoing investigations, is built on the assumption, That the hydrostatic pressure at every point of the fluid is the same function of the co-ordinates of the point. The accelerating forces are represented by the partial differential coefficients of the pressure ; and therefore they are likewise the same functions of the co-ordinates of their point of action in every part of the mass. The whole reasoning rests on these fundamental points ; and if the state of a fluid were such that they are not verified, the equations for determining the required figure could not be formed, and the equilibrium would be impossible. As the hydrostatic pressure is known only by means of the given accelerating forces, it seems most suitable to em- ploy the properties of the latter in laying down what is required for the equi- librium of a mass of fluid. It is necessary, and it is sufficient for the equili- brium of a homogeneous fluid, first, that the accelerating forces acting in the directions of the co-ordinates be, in every part of the mass, the same functions of the co-ordinates ; and, secondly, that these functions possess the conditions of integrability. When these two conditions are both fulfilled, the determina- tion of the figure of equilibrium is reduced to a question purely mathematical. For we can form the equation (1) which makes the accelerating forces balance the variation of pressure ; and, by integrating this equation, we obtain the hy- drostatic pressure, from which is deduced the equation of all those points at which there is no pressure, or in other words, the equation of all those parts of the outer surface which are at liberty. Nothing more is required for se- curing the permanence of the figure of the fluid, except that the pressures pro- pagated through the mass be either supported or mutually balance one another. The conditions for the equilibrium of a homogeneous fluid, as they are here laid down, do not enable us in all cases to form immediately the equation of the figure of equilibrium. If the particles attract or repel one another, the accelerating forces will, for the most part, vary as the fluid changes its form, AND THE FIGURE OF A HOMOGENEOUS PLANET. 119 and they may not be at eveiy point the same functions of the co-ordinates in all the figures, of which it is susceptible ; but, notwithstanding the equilibrium may still be possible, because this indispensable condition may be fulfilled when figures of a certain class are induced on the mass. In such cases, the deter- mination of the equilibrium necessarily requires two distinct researches ; of which one is to find out what are the particular figures into which the mass must be moulded, so as to make the accelerating forces at every point the same functions of the co-ordinates. After these figures have been found, we can apply to them the equations expressing the conditions of equilibrium, and ac- complish the mathematical solution of the problem. But if it shall appear that no figure whatever capable of fulfilling both the conditions laid down above can be induced on the fluid, the equilibrium will be absolutely impossible. In the usual exposition of this theory, the equilibrium is made to depend on conditions that do not exactly coincide with those at which we have arrived. According to Clairaut and all other authors who have written on this sub- ject, it is necessary, and it is sufficient, for the equilibrium of a homogeneous fluid, first, that the expressions of the accelerating forces possess the criterion of integrability ; secondly, that the resultant of the forces in action at all the parts of the outer surface which are at liberty, be directed perpendicularly to- wards these surfaces. We may throw out of view what regards the criterion of integrability, about which there is no difference of opinion, and which in reality is always fulfilled by the forces that occur in physical researches. The perpendicularity of the forces to the outer surface is a property of the differen- tial equation of that surface, and will necessarily take place whenever it is pos- sible to form that equation. Nothing more is required for forming the equa- tion mentioned, than that the accelerating forces at every point of it be ex- pressed by the same functions of the co-ordinates of the point.* It follows * The forces are perpendicular to every surface in which the pressure is constant. The outer sur- faces are those at every point of which there is no pressure. In all the questions that have occurred, the forces at the outer surface of the fluid are the same functions of the co-ordinates of the point, what- ever geometrical figure the fluid is supposed to assume ; and on this account the equation of the outer surface can be formed without reference to any particular class of figures. But this is not sufficient ; for, according to the fundamental assumption laid down by Clairaut himself, the theory of equilibrium cannot be applied, unless the forces be the same functions of the co-ordinates of their point of action in every part of the mass. 120 MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, therefore, that the difference between the conditions of equilibrium hitherto universally adopted, and those laid down above, amounts to this : according to the former it is required that the expressions of the accelerating forces be the same functions of the co-ordinates at every point of the outer surface, this being all that is necessary for forming the differential equation of that surface; according to the latter, the forces will not balance the pressure, and the laws of equilibrium will not be fulfilled unless the forces be the same functions of the co-ordinates at every point whether situated in the outer surface, or in the interior part of the mass. If a homogeneous fluid, of which the particles are urged by accelerating forces be in equilibrium, all that is required by Clairaut’s theory will un- doubtedly be fulfilled ; but the converse of this cannot be affirmed. It is no where proved generally by unexceptionable arguments, and indeed no proof can possibly be given, that the forces in the interior parts of the fluid will balance the pressure, merely because the resultant of the forces in action at the outer surface is perpendicular to that surface. All the attempts that have been made to demonstrate this point, tacitly assume that the expression of the forces is the same at the surface and in all the interior parts ; which is not uni- versally true. In a very extensive class of problems the difference between the two ways of laying down the conditions of equilibrium disappears. This will happen when the accelerating forces are independent of the figure of the fluid, as will be the case if the particles exert no action on one another by attraction or repulsion. In such problems the forces impressed upon every particle, what- ever be its situation, and whatever be the figure of the fluid, are by the hypo- thesis, the same given functions of the co-ordinates. The figure of equilibrium will be the same whether, following Clairaut, we obtain the equation of the outer surface by means of the forces in action at that surface, or, making use of the property that the pressure vanishes at all the points where the fluid is at liberty, we deduce the same equation from the pressure that prevails generally throughout the mass. But Clairaut’s theory cannot be extended to the solution of other problems than those of which we have been speaking. In no other cases is it evident without inquiry that the proposed accelerating forces urging a particle, are, in AND THE FIGURE OF A HOMOGENEOUS PLANET. 121 every part of the mass, the same functions of the coordinates of the particle ; and unless this be verified, the theory of equilibrium cannot be applied. In a homogeneous planet in a fluid state, there are forces which prevail in the in- terior parts and vanish at the surface ; and, as Clairaut’s theory notices no forces except those in action at the surface, it leaves out some of the causes tending to change the figure of the fluid, and therefore it cannot lead to an exact determination of the equilibrium. II. Application of the foregoing Theory to the Question of the Figure of the Planets. 7. Having now explained the general theory of the equilibrium of fluids at sufficient length, I proceed to apply it to the question of the figure of the planets, in which it is required to determine the equilibrium of a fluid entirely at liberty, and unconfined by any obstacle or support. The problem is one of considerable difficulty. It is necessary to distribute the investigation under distinct heads. It would otherwise be impossible to preserve perspicuity and precision of ideas in an inquiry essentially different in different hypotheses. The equilibrium of a homogeneous fluid must occupy our attention before that of one having its density variable. For although it may at first appear that the latter problem is the more general, and includes the former, yet it will be found that the equilibrium of a fluid of variable density, depends upon that of a homogeneous fluid, and is deducible from it. And even with regard to homogeneous fluids, distinctions must be made, because what is required for the equilibrium varies with the nature of the accelerating forces. In this respect we distinguish these two general cases, of which we shall treat in two separate problems ; First, when the accelerating forces depend only on the co- ordinates of their point of action, and are explicitly known when the coordi- nates are given ; Secondly, when the accelerating forces depend not only upon * the coordinates of the particle on which they act, but likewise upon the figure of the whole mass of fluid ; as happens for the most part when the particles attract or repel one another. MDCCCXXXI. R 122 MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, Problem 1st. — To determine the equilibrium of a homogeneous mass of fluid which is entirely at liberty, when the accelerating forces are known func- tions of the coordinates of their point of action. The equilibrium of a mass of fluid which is entirely at liberty, can depend only upon the action of such forces as tend to change the relative position of the particles with respect to one another. It is not affected by any motion common to all the particles, nor by any force which acts upon them all with the same intensity in the same direction ; the effect of such motion, or of such force, being to displace the centre of gravity of the whole mass without altering the relative situation of the particles. In estimating the accelerating forces upon which the figure of equilibrium will depend, we must therefore begin with reducing the centre of gravity, if it be in motion or urged by any force, to a state of relative rest ; which is accomplished by applying to every particle a force that would cause it to move with the same velocity as the centre of gravity, but in a contrary direction. In the investigation of this problem we may therefore suppose that the centre of gravity is at rest and undisturbed by the action of any accelerating force. Suppose now that a mass of homogeneous fluid entirely at liberty, is in equi- librium, and conceive three planes intersecting at right angles in the centre of gravity of the mass, to which planes the particles of the fluid are to be referred by rectangular coordinates. Let x, y, z , represent the coordinates of a particle, and having resolved the accelerating forces acting upon it into other forces that have their directions parallel to the coordinates, put X, Y, Z, for the sums of the resolved parts respectively parallel to x, y, z, and tending to shorten these lines. According to the hypothesis of this problem, the forces X, Y, Z, depend only upon the coordinates of their point of action ; and they are at every point the same functions of those coordinates. The equilibrium will therefore be impossible unless X d x + Y dy + Zc/z be an exact differential, this being necessary in order that the hydrostatic pressure be a function of three independent variables as the fundamental assumption of the theory demands. Let

= (Bj x 2 + B 2 y 2 + B 3 z 2 + V>±xy + B 5 xz + B 6 yz) + (Dj x? + D 2 y 3 + D 3 2 3 + D 4 x 2 y + &c.) 4- &c. That the expression of

i 3 + D 3 + D ! 2 !j + &c.) + &c. The symbols |, r h represent three rectangular coordinates of a point in the surface of a sphere having unit for its radius ; and, in order to simplify, I shall write Q 2 , Q 3 , and generally Q„, for homogeneous functions of g, y, of two, three, and n dimensions : then,

z = gZ‘ The equations of equilibrium will, therefore, be

— h 2 sin 2 4/ dtp V 1 — k 2 sin 2

4), which itself remains indeterminate. We may therefore suppose p X K (X x ) = K (y)> P representing any integer number; and, in conse- quence, we shall have K(X.) = j K, K(A 2 ) = J-K, ... K(X„) = ~K. Any proposed number being assumed for p, we may determine the amplitudes X 1} X 2 , ?. 3 , &c. by the theory for the multiplication and subdivision of elliptic functions : but as the equations to be solved are complicated and impracticable, the arcs X l5 X 2 , &c. may be treated as known quantities without any attempt to compute them. An elliptic function becomes equal to the arc of its amplitude, when the modulus vanishes : and in this case the arcs X l5 X 2 , X 3 , &c. are obtained by the 1 7T subdivision of the quadrant of the circle, and are respectively equal to — . — , 2 7T 3 TT o ~p * ~p • &C ' Having made these observations, we shall for the present dismiss all consi- deration of the equation to be demonstrated, and turn our attention to inves- tigate two variable arcs ^ and ); then, as u is a variable quantity depending upon the amplitude according as (2 n + 1) u holds an odd or an even rank in the series of the odd multiples of a. On the other hand, Avhen u is zero, or equal to 2 na any even multiple of a, we shall have y = 0, one of the factors of the numerator necessarily vanishing; for in a sequence of the even multiples of co, of which the number is p, there must be one equal to 2 pco, or to a multiple of 2 pco; and therefore when u — 2 n co, one of the factors must be the sine of an amplitude equal to 1 80° or to a multiple of 180°. Further, let co — z be substituted for u in the expression (B), a being less than a ; then, sin A (co — s) sin A (3 co — s) .... sin A (2 p co — co — z) y — y Now, in the numerator, the partial products, of the first and last factors, of the second and last but one, and so on, are as follows : sin A (co — 2) sin A (2p co — co — z) = sin A (a — z) sin A (co -j- z), sin A (3 co — z) sin A (2 p 00 — 3 a — z) = sin A (3 u — z) sin A (3 a + z), &c. to which we must add the single factor sin A (p a — z), when p is an odd num- ber. All the partial products, it will be observed, have the same value whe- ther z be positive or negative; and they are all greatest, when £ = 0, as will readily appear from what is proved in § 2. Wherefore y has the same value and the same sign, when u is at equal distances from the limits 0 and 2 a ; and it attains its greatest magnitude, equal to 1, when u = oo. And, if we substi- tute (2m+ 1) co — z for u, this substitution will not change the foregoing factors, but only their order, and the sign of their product, which sign, while u is contained between the limits 2 nu and 2w»-)-2«, will be + or — , accord- ing as (2 11 + 1 ) co holds an odd or an even rank in the series of the odd mul- tiples of u. We may now conclude, from what has been proved, that y , in the expression (B), represents the sine of an arc ' = (3 z sin 8 A 2 1 - sin 2 A. 1 - sin 2 Ap _ 1 1 — k 2 z* sin 2 A 2 " 1 — Jc 2 z 2 sin 2 A 4 ’ ’ * 1 — /c 2 2 2 sin 2 Ap — 1 * ( 2 ) When p is an even number, if we leave out the first factor in the numerator of the expression of y or sin there will remain an odd number of factors, that which occupies the middle place, being sin A (u -j- p co ) : and any factor, as sin A (u + 2 nu), between the first and the middle one, will have another, viz. sinA(«< + 2 poo — 2 nu), corresponding to it after the middle one; and the product of this pair of factors will be obtained as before, viz. sin A (u + 2 n a) sin A {u + 2 p a — 2 n a) = sin 5 A 2w - sin 2

) ; and sin A {u + p u) = cos V 1 — k 2 sin 2

• ( 5 ) In like manner, if P and R represent the rational binomial products in the numerator and denominator of the formula (3), we shall have and • , 3 z */ 1 — z 2 P sl ^- 7r-tv x B 2 , _ (1 -£ 2 z 2 )R 2 -/3 2 z 2 (l — z°~) P z T t 1 7,2 ~‘2\ T? 2 (1 — lc 2 z 2 ) R 2 Proceeding as before, it will appear that the numerator of this expression is 3 A MDCCCXXXI. 360 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 2 2 divisible by the double divisor ^1 — ^ , 2 n + 1 being any odd num- ber less than p ; and in this case when p is an even number, all the divisors are double. Wherefore the product / 1 _ y ( i fi-Y (\ A—) 2 (\ — y ^ sm-x l J * V sin-Ag/ ' y sin 2 A 5 / * ' * * y sin 2 X p _ i / will divide the numerator of the value of cos 2 ^ ; and it will be identical to it, because both the expressions have the same dimensions. Thus we obtain, p being an even number, l - sin 2 Aj 1 — s 2 sin ' 2 A, 1 - sin 2 A p — i cos

sm ? = z: and the differential of equation (1) will become by substitution. ( 1 — k 2 z~) R 2 J / z V 1 — z 2 . P \ dz ' Va/i -f-^Tr/ sj ^(1 — k° %•) R 2 — (3 2 z 2 (1 — z- ) P 2 ^ ^(1 — k 2 z 2 ) R 3 — /3 2 h 2 z 2 (l — z 2 ) P 2 ^ = — t-—: o 1 . n : and, on account of the formulas (D), Vl — z 2 .l - k 2 z 2 ’ v (i -F~z 2 ) R a A / z . P \ dz c * Wi_ k 2 z 2 . R/ Q.R' I V 1 — z 2 . 1 — k 2 z 2 It will be proved, by the like reasoning as before, that the numerator of the left side of this equation is divisible by the product in the denominator. Now if we perform the differentiation indicated, we shall find, S = (1 - 2* 2 + k 2 z 4 ) PR+ * (1 - z 2 ) (1 - k 2 z 2 ) R 2 (1 - FF) R 2 / z >/i - dz ' \V\ — Jc 2 z 2 * R/ — and it is evident that all the rational factors of the left side of this last formula, and consequently all the factors of Q X IV, will be factors of S. By substi- tution the differential equation (1) will now become _s ... _ 1 Q x R' ~ ’ which is manifestly verified : for, as Q X IV divides S, and the two expressions MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 365 have the same dimensions and the same absolute term, they are identical. The equation (1) is therefore demonstrated when p is an even number. 7- The transformation expressed by the equation, /*’#• d-\> ^ d tp J 0 V 1 — k 2 sin® \|/ J Q a/ 1 — k' 2 sin® — 1 1 + tc V — 1 1 + fxy / -l \1 + c s x / and from this we deduce, p being an even number, 4 1 = V + \$ + ) approach to a cir- cular arc as near as may be required. If we wish to apply the same theorem to reduce the given function F ( h , cp) to a logarithm, through a scale of increasing moduli, the process is not so direct. For, in the first place, the greater modulus k is not immediately de- ducible from the less h, by means of the formulas that have been investigated; and, in the second place, the amplitude

£,# = tan 1 - y = (3x X 1 — will have for its solution. MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 369 X- X 2 X 2 1 -f- 7 TT~ 1 “p 7 o~T~ 1 "I - 7 5~7 _ o x x tan-A^ _ tan- A 4 . . . tan-^-i ^ y ' x 2 x 2 1 tan 2 Aj 1 tan 2 A 3 * tan 2 A P - \ In this equation the values of y and x are between 0 and + 1, which limits they both attain at the same time. If we make x = + 1, and attend to the value of (3, we shall find?/ = + 1. Let y = sinT, x = sin), (15) the letters 4 and sin 3 A;, _ i sin 2 yp sin r = |8' sin ^ X tan „ 1 , , sin* \f/ 1 + tan 3 1 + 1 + sin 2 \{/ tan 3 fj. p - i sin 2 yp tan 3 j up - 2 When a multiple is required, we pass directly, by means of the two equa- tions, from the given amplitude (p to r which is sought. In the case of an aliquot part, the amplitude r being given, the solution of the second equation, of which p is the dimensions, will determine sin ^ ; and the amplitude (p which is sought, will then be found by solving the first equation, which is also of p dimensions. From the nature of the second equation, it has only one real root, and p — 1 impossible roots, for every real value of sin r ; and therefore it follows from the first equation, that the amplitude

1 - tan 3

and ,+) = (1 + U) F (A,

~ f TV’ &C ‘ : and, if we likewise deduce a series of amplitudes in this manner, tan (\p — 'M f (£,?>) = t • i T F ( /i i>+i) F (A,p) = T • T • T* • F (h 2 , 'W, &c- In the second case, when we would pass from the less modulus h to the greater Jc, the amplitude

k 2 — 1 + V ’ and further, if the amplitudes ip, &c. be deduced from the formulas, sin (2 (p — ip) = h sin ip, sin (2

-of the sun. n . . mean motion in its orbit u . . obliquity of the ecliptic L v n t -j- zj — yp r

= — - — + — - — + — - — cos 2 0—^ — - — — — - — > cos 2 5 + — | cos (2 0 — 2 S) + cos (2 0 + 2 5) j + s !IlJL£ s in 2 S cos 0. 3 d 2 38G MR. LUBBOCK ON THE TIDES IN THE PORT OF LONDON. If the longitude of the sun be introduced by putting for § its value from the equation sin o = sin w sin l l being reckoned from the first point of Aries, { sin

cos 2 l 14 2 / + S ‘ n -- — sin 2 S cos 0 Considering the results of many years so as to destroy the effects of changes in the moon’s parallax and declination, and taking the mean of the times of high water when the moon passes the meridian at any given time, and twelve hours later, the tides, owing to the united action of the sun and moon, depend on the terms 3 m II 3 cos 2 p 4 " cos 2 0 + + 3 mil 3 cos q