^ray 17, 1 877] 



NA TURE 



49 



clianism, especially in " parallel motions," both real and ap- 

 proximate. 



Tlie familiar theorem that the relative velocities of points in any 

 body vary as their instantaneous radii needs merely to be men- 

 tioned. It is to be regretted that it is not more generally used, 

 for while it does not increase the difficulty of comprehendinj; 

 simple cases, it is of enormous advantage in simplifying such 

 (apparently) complex ones as not unfrequently occur in me- 

 chanism. 



riie expression for static equilibrium is also tolerably fami- 

 liar : — the sum of the moments of all the forces actini^ upon a body 

 about its inst. centre must = o. For practical purposes, how- 

 ever, it is generally more convenient to state the proposition : — 

 the resultant of all the forces acting -upon a body must pais 

 through the foint of contact of its centroiJs. The application 

 of this proposition to all the simpler cases is self evident, and at 

 the same time it reduces complex cases to their smallest possible 

 dimensions, rendering most very easy, and in many cases ijreatly 

 aiding the comprehension of the alterations in conditions of 

 equilibrium corresponding to consecutive alterations in the 

 positions of mechanisms as their links move. It may just be 

 noted that as the two forces of a couple have for their resultant 

 a force (infinitely small) acting along the line at infinity, the 

 jiroposition gives at once that where the inst. centre of a body 

 is at infinity it is in equilibrio under any number of couples of 

 any magnitude. In the case of a body moving parallel to itself, 

 therefore (see ante) all couples may be neglected so far as its 

 static equilibrium is concerned, whatever their magnLtufle or 

 sense. 



The following are, in conclusion, a few of the kinetic pro- 

 positions the solution of which is greatly aided by the use of 

 centroids : — 



(I.) If a force' constant in direction and position act upon a 

 body, then (i.) if it cut the centroid for the motion of the body 

 in one or more points motion will take place until the first of 

 these becomes the point of contact, and will then cease ; (ii.) if 

 it pass entirely without this centroid, there will be continuous 

 motion. As corollaries to (i) may be mentioned (a), if the cen- 

 troid for the motion of a body be a curve of the 2nd, 3rd . . . Hth 

 order, the body has a maximum of 2, 3 ... ?; positions of equi- 

 librium under some one or more forces constant in direction and 

 position. Also (/'), if a body have not more than a single 

 position of equilibrium under any such force, the centroid for its 

 motion must be a straight line. 



(II.) If the position of a force relatively to the body upon which 

 it acts remain constant, then (i.) if it cut the centroid of the body 

 in one or more points, motion will take place until one of these 

 becomes the point of contact, (ii. ) if it lie entirely without the 

 centroid of the body, there will be continuous motion. This 

 gives corollaries as to positions of equilibrium similar to those 

 just stated. 



(III.) If a force constant in direction act always at the same 

 point of a body, motion will continue until the instantaneous 

 radius of the pomt becomes parallel to the direction of the force. 

 There is here no case of continual motion ; the theorems as to 

 number, &c., of positions of equilibrium are similar to those given 

 above. 



English writers have used these curves very little. Among 

 modern continental writers who have employed them may be 

 mentioned Dwelshauvers-Dery (Liege) who uses them in his 

 " Cinematique " for questions relating to relative velocities ; Schell 

 (Carlsruhe) in his "Theorie der Bewegung u. d. Kralte ; Reu- 

 leaux (" Theoretische Kinematik," and elsewhere), who gave 

 them the name [Polbahnen), by which they are known iir Ger- 

 many, and who has used them ably and extensively for kinematic 

 problems ; and lastly Proll, who has made use of them in his 

 recent " Versuch einer graphischen Dynamik." The writer has 

 not, however, found them anywhere unreservedly adopted, and 

 has, therefore, made this attempt to show how easily cenlroidal 

 methods adapt themselves to the general treatment of mechanical 

 problems, especially those connected with mechanism, and at the 

 same time how well suited they appear to be for educational 

 purposes. 



OUR ASTRONOMICAL COLUMN 



The Total Solar Eclu'se of 1SS2, May 17. — Hallaschka, 



in his " Elementa Eclipsium," describes this eclipse as broadly 



total, whereas, it will be, in reality, total, though the zone of 



* Or here, and in the following propositions, the resultant of any number 



of forces. 



totality will not be a broad one. An error in the moon's semi- 

 diameter led to the statement in Hallaschka's work. The 

 following elements of this eclipse, calculated upon the same 

 system that has been applied in the examination of other solar 

 eclipses in this column, will probably be near the truth : — 

 Conjunction in R.A., May 16, at igh. 41m. Ii'7s. G.M.T. 



R.A 



Moon's hourly motion in R.A. 

 Sun's „ ,, „ 



Moon's declination 

 Sun's „ 



Moon's hourly motion in deck 

 .Sun's ,, „ ,, 



Moon's horizontal parallax . . . 

 Sun's ,, ,, 



Moon's true semi-diameter ... 

 Sun's ,, ,, 



53 56 35-4 

 36 I4'S 

 2 2S7 

 19 3S 46-4 N. 

 19 19 38-8 N. 

 4 56-0 N. 

 o 33-8 N. 

 58 I5-I 

 8-8 

 15 52-4 

 15 48-8 



The central .and total eclipse begins at i^h. 53 Sm. in longitude 

 3° 11' W., and latitude 10° 40' N. ; it occurs with the sun on 

 the meridian in 63° 44' E., and 38' 35' N., and ends at 

 21I1. i8'8m. in 138° 51' E., and 25° 25' N. The following are 

 points upon the central line in that portion of its track where 

 observations are most likely to be made : — 



The central line therefore commences in the west of Africa, 

 and traversing that continent in the direction of Upper Egypt, it 

 passes over the Nile below Thebes, thence over the extremity of 

 the peninsula of Sinai, near Ras Muhammed, and almost directly 

 over Hillah, the site of the ruins of Babylon, to Teheran. The 

 position of this capital according to Gen.j Stebitzky (Astron. 

 Nach., No. 2,113) is in longitude 3h. 25m. 417s. E. of Green- 

 wich, and latitude 35° 41' 7", this point referring to the station 

 of the Indo-European telegraph ; so that the central line of 

 shadow according to our elements passes sixteen miles to the 

 south of it. Calculating directly for this longitude we have the 

 following results : — 



h. m. s. 



Totality begins May 16 at 22 36 29 ) , , ^ „ 

 „ ends „ 22 38 13 jlocatM.i. 



Duration I 44 



The sun at an altitude of 67°. So that the greatest duration 

 of totality in _this eclipse about 12° east of Teheran is about 

 im. 45s. 



The central line subsequently traverses China, passing off at 

 or close to Shanghai, at which place a _total eclipse of short 

 duration may be observed. 



The next total solar eclipse on July 29, 1878, which crosses 

 the United States is pretty fully noticed in the various Epheme- 

 rides, though in due .time the American astronomers will no 

 doubt provide a chart showing on a larger scale the breadth and 

 position of the zone of totality over their country. Then follows 

 the total ecUpse of J.anuary 11, 1S80, in which the track of the 

 central line lies almost wholly upon the Pacific, the total phase 

 being visible for a brief duration only near the coast of California, 

 above San Francisco. The total eclipse of May, 1 882, of which 

 the elements are here given is the next in order of date. 



The Comets of 1402. — It is singular, considering the atten- 

 tion which the Chinese paid to the observation of comets, their 

 annals containing reference to several hundreds of these bodies, 

 should not have recorded the appearance of the two evidently 

 great comets of 1402. In particular is this the case with the 

 first comet, which, according to the descriptions in the European 



