SCIENCE. 



319 



scend farther than half way to Uranus, or 500,000,000 

 miles ; then all the matter alluded to in this paper will be 

 above the chord. So long as the mass remains spherical 

 the chord will cut out a segment of a circle. Now let the 

 cosmic sphere receive some unknown impulse that will 

 accelerate its velocity of rotation, and the mass will 

 change to a spheroidal form. The chord of the arc will 

 shorten, matter at the equator will become elevated and 

 sections of the protuberance will change curvature. Make 

 both ends of the chord points of tangency, and produce 

 tangents to the curve to infinite space. Then if rotation 

 accelerates, the curve bounding the ascending equatorial 

 protuberance must continually change form, and the tan- 

 gents, direction ; while sectional curves will pass all var- 

 ieties of the hyperbola, parabola and ellipse. Thus let 

 the mass become very oblate, pass a cutting plane down 

 to the chord, and the curve cut out will be hyperbolic. 

 Increase rotation; the equator will become higher, the 

 chord shorter and the sections parabolic. Let the veloc- 

 ity be still accelerated, the equatorial matter will be lifted 

 to greater altitudes, the chotd will be shorter than ever, 

 and the sections elliptical. To this reasoning the objec- 

 tion may be raised by some that no matter how flattened 

 the mass might become, sections cut to the chord of the 

 arc would in every case be elliptical. We do not insist 

 that they would be hyperbolas or parabolas ; but will 

 prove if ellipses, that elliptical segments are more fatal 

 to the theory of ring formation than are segments of any 

 other form of curve. Two factors engaged in the evolu- 

 tion of cosmic rings — gravity and an opposing force gen- 

 erated by rotary motion. We attack the whole Nebular 

 Hypothesis with the fact that if the revolving gaseous 

 mass abandoned matter at present existing in Neptune, 

 the planet is now in the position of the centre of gravity 

 of the detached portion. This being true, Neptune 

 never became a member of the solar family by displace- 

 ment ot its material from the original mass, because no 

 mass could have been cast off whose centre of gravity 

 coincided with the orbit of that world. Let us see if the 

 Neptunian ring was abandoned when the cosmic mass 

 was a sphere. If sp, the ring was either cylindrical or 

 a segment of a circle. But the centre of gravity of a sec- 

 tion of a cylindric ring is in the centre of the section. 

 Since Neptune now traverses a path once the centre of 

 gravity of the ring, it follows that when detached the 

 spere of gas was larger than a ball bounded by the Nep- 

 tunian orbit, as there must have been as much matter 

 above the centre ot a section of the ring as below. The 

 larger the sphere the slower the rotation, hence it did 

 not rotate as rapidly as it would, had it been equal in size 

 to a globe having the diameter , of Neptune's track. 

 But it had *o revolve faster to detach a ring because 

 Neptune now moves on an orbit with a velocity of 3.36 

 miles per second ; yet displays no tendency to leave it on 

 a tangent. And greater detaching force would have 

 been required to cause a ring to leave the equator than 

 would now be necessary to throw Neptune off its orbit, 

 because the force had to overcome what little cohesion 

 the dissociated atoms had. The sphere must have been * 

 far larger than the path of Neptune, because the ring, be- 

 ing abandoned at the equator, had to be hundreds of 

 millions in thickness to secure gas enough to condense 

 into the planet, and its rate of rotation proporlionately less 

 than its present velocity. 



It is certain that the ring whence Neptune was formed 

 was not cylindrical. The only other possible form of 

 ring is segmental. The distance of centres of gravity of 

 all circular segments from the centre of the circle can be 

 calculated. The problem resolved itself into this : — given 

 the distance of the centre of gravity of the segment of a 

 circle from the centre, to find the dimensions of the seg- 

 ment, and radius of the circle. We know that Neptune 

 is in the position of the centre of gravity of whatever 

 shaped mass was detached. But it lies on the circum- 

 ference of a circle whose radius is the distance to the 



sun. Therefore the circle must have been larger than its 

 orbit to be able to afford a segment having sufficient size 

 to have its centre of gravity coincide with the track of 

 Neptune. In all these computations we take the distance 

 of Neptune from the sun to be 2,780,000,000 miles. — Ele- 

 ments of 1850, Newcomb's Astronomy. The ring of 

 whatever shape is supposed to be detached, severed, 

 straightened, and cut into an infinite number of sections 

 perpendicular to its length. In the case in question, 

 sections are segments of a circle, and we are in search of 

 the radius of the circle whence the segment was cut. 

 We have found the length of the radius to be 3,000,000,- 



O 



000 miles, by means of the formula, G = — ^ wherein 



G=the distance of the centre of gravity of the segment 

 from the centre of the circle. 



C=the chord of the arc, or base of the segment. 

 A=the area of the segment. 



That is — " Divide the cube of the chord of the segment by 

 twelve times the area of the segment ; the quotient will 

 be the distance of the centre of gravity required from the 

 centre of the circle." — Vogde's Mensuration p, 237. 

 Making approximation with a circle whose radius was 

 2,900,000,000 miles, with chords at different distances 

 within the Neptunian orbit, it was found in two trials that 

 a circle of that radius was untenable. Using a circle 

 having a radius of 3,000,000,000 miles, and chord de- 

 scending 300,000,000 miles, it was soon found that the 

 centre of gravity of that segment was in distance from the 

 centre equal to the distance of Neptune from the sun. 

 But the chord was 2,600,000,000 miles long ! Does any- 

 body believe that a break took place along a line of such 

 length, and 300,000,000 miles below the equator of the 

 sphere ? Was detachment possible when the sphere ro- 

 tated slower than the orbital velocity of Neptune now is, 

 yet shows no signs of elevating to a tangent to its path, 

 though moving with unimpeded force ? The first world 

 was not abandoned by the cosmical mass when a sphere. 



Could it have been formed from the matter contained 

 in the segment of any other curve known to geometers ? 



To find the centre of gravity of a parabolic area : — ■ 

 " The centre of gravity is on the axis, at a distance from 

 the vertex equal to three-fifths the altitude of the seg- 

 ment." Peck s Calculus p. 175. Then Neptune, as it is 

 the centre of gravity of the parabola must be two-fifths 

 above the base or limiting plane of the curve, We have 

 made calculation of the altitudes of several possible para- 

 bolas, by locating the base at different distances between 

 the orbits of Uranus and Neptune. The following table 

 shows the distances of the limiting planes below Neptune, 

 the altitudes of the segments, above the base, — above 

 Neptune, — and also gives the diameter of the mass on the 

 hypothesis, that it could have been so elongated as to make 

 it possible that parabolas could be cut out of the equator 

 by perpendicular planes. 



Table I. Altitudes of Parabolas. Distances in Miles. 



Distances of 

 Base 

 Below Neptune. 



Altitudes 

 Above 

 Base. 



Altitudes 

 Above 

 Neptune. 



Diameters of 

 Mass when so 

 Expanded. 



500,000,000 

 400,000,000 

 300000, 000 

 200, 000, coo 

 100,000,000 

 50,000,000 

 25,000,000 



1,250,000,000 

 1,000,000,000 

 750,000,000 

 500,000,000 

 250,000.000 

 125,000,000 

 62,500,000 



750,000,000 

 600,000,000 

 450.000,000 

 300,000,000 

 150,000,000 

 75,000,000 

 37,500,000 



7,060,000,000 

 6,760,000,000 

 6,460,000,000 

 6,160,000,000 

 5,860 000,000 

 5,710,000,000 

 5,635,000,000 



Should these figures be deemed unsatisfactory, be- 

 cause they relate to sections or surfaces, while actually 

 considering a solid ring, a table of paraboloids is in- 

 serted. The ring was 17,467,000.000 miles long, cut it 

 in an infinite number of parabolic sections ; revolve each 



