EAR
222
ear
the author’s conclusions. In the second part of that treatise, it isproveil, that putting p for the polar force, n for the < equatorial;
3 for the true ellipticity of the earth’s figure, and , for the ellipucityof the homogeneous spheroid.
F -Z'.I r 2 « — S. therefore 3 2« — ---
it n
and therefore, according to your observation, J = j4- r . This is thejust conclusion from your observations of the pendulum, taking itfor granted, that the meridians are ellipses: which is an hypothesis,upon which all the reasonings of theory have hitherto proceeded.Rut plausible as it may seem, I must say, that there is much reasonfrom experiment to call it in question. If it were true, the incre-ment of the force which actuates the pendulum, as we approachthe poles, should be as the square ofthe sine of the latitude, or, which,is the same thing, the decrement, as we approach the equator,should be as the square of the cosine ofthe latitude. Rut whoevertakes the pains to compare together such of the observations of thependulum in different latitudes, as seem to have been made withthe'greatest care, will find that the increments and decrements dobv no means follow these proportions; and in those which I haveexamined, l find a regularity in the deviation which little resemblesthe mere error of observation. The unavoidable conclusion is,that the true figure of the meridians is not elliptical. If the me-ridians are not ellipses, the difference ofthe diameters may indeed,or it may not, be proportional to the difference between the polarand the equatorial force ; but it is quite an uncertainty, what re-lation subsists between the one quantity and the oilier; our whole jtheory, except so far as it relates to the homogeneous spheroid, isbuilt upon false assumptions, and there is no saying what figure ofthe earth any observations of the pendulum give.” Dr. Horsleythen lays down the following table, which shews the different re-sults of observations made in different latitudes; in which the firstthree columns contain the names of the observers, the places ofobservation, and the latitude of each; the fourth column shews thequantity of p —ii in such parts as ri is 100000, as deduced fromcomparing the length of the pendulum at each place of observa-tion, with the length ofthe equatorial pendulum a*termed by M.Bouguer, u|>on the supposition that the increments and decrementsof force, as the latitude is increased or lowered, observe the pro- Iportion which theory assigns. Only the second and the last valueofp — n are concluded from comparisons with tiie pendulum atGreenwich and at London , not at the equator. The fifth columnshews the value of 3 corresponding to every value of v — u, ac-cording to Clairault’s theorem: * !
Observers.
Places.
Lat.
Equator.
0°
0’
1
Porto Bello.
9
34
741.S
Green.
Otaheitoe.
17
oo
5G3.2
Bouguer.
San Domingo.
18
27
591.0
:
Abbe de la Cable
Capeol Good Hope
33
55
731.5
48
50
I
The Academicians
Pello.
66
48
5G5.9
I
ys?
Captain Phipps.
1 79
50
471.2
ITT |
" Bv this table it appears, that the observations in the middle jparts of the globe, setting aside the single one at the Cape, are as jconsistent as could reasonably be expected; and they represent ithe cllipticitv of the earth as about one 3401 h. Rut when we comewithin.ten degrees of the equator, it should seem that tiie force of |gravity suddenly becomes much less, and within tiie like distance jof the poles much greater than it could be in such a spheroid.” iThe following problem communicated by Dr. Leather!,nul to Dr.Pemberton, and published by Mr. Robertson, serves to find tiieproportion between the axis and the equatorial diameter, frommeasures of a degree of the meridian in two different latitudes sup- tposing the earth an oblate spheroid. Let AYap (Plate LIX. fig. j15) be an ellipse representing a section of tiie earth through the |axis Yp ; the equatorial diameter, or the greater axis of tin* el’ipse,being A a ; h’l L and P lie two places, where the measure of a de-gree lias been taken; these measures are proportional to the radii Iof curvature in the ellipse at those places; and if CQ, C’R, he con- jj ligates tothe diameters whose vertices are E and F, CQ will lie to jCR in the subtriplicated ratio of the radius of curvature at K to j
that at F, by cor. 1, prop. 4, part G, of M lines’s Come Sections,and therefore in a given ratio to one another; also the angle. , QC1’,RC P, are the latitudes of E and F: these angles are Iheiemre given,as w ell as the ratio of CQ, to CR. Let QVand R-5 lie drawn paral-lel to Pp, ami QYW to An; then from the properly of the ellipsethere is given tiie ratio of YC 3 —ZC 3 to KZ 3 — <i\ J , this ratiobeing tiie same with the ratio cf CA 3 to CP 3 ; tiiereiore tiieratio of CA to CP is given, lienee, if the sine and cosineof the greater latitude be each augmented in the subtriplicateratio of the measure of the degree in the greater latitude tothat in tiie less, then the difference of the squares of tiie augment-ed sine, and the sine of the less latitude, will be to the differenceof the squares of the cosine ot the less latitude, and tiie augment-ed cosine, in tiie duplicate ratio of tiie equatorial to the polardiameter. For C q being taken in CQ equal to CR, and qu drawnparallel to QY, C u and uq, CZ and ZU, w ill lie the sines ami co-sines of the respective latitudes to the same radius; and CV, VQ,will be the augmentations oiCuand C q in tiie ratio named. lienee,to find the ratio between the tw o axes of tiie earth, let F, denote thegreater, and F tiie less of tiie two latitudes, M and N the re-spective measures taken in each; and
let Y denote 3
: then
cos.sp _ ]>■> y cos. 3 E less a xis
■\ l’ 3 X sin. 5 F. — sin. 3 F 1S greater axis
it also appears from tiie above problem, that when one of the de-grees measured is at the equator, the cosine ol the latitude of theother being augmented in the subtriplicate ratio of the degrees, thetangent oi the latitude will be to the tangent answering to the aug-mented cosine, in tiie ratio of the greater axis to the less. For,supposing F. the plac-* out of the equator, then, it tiie semi-circleYlmnp lie described, and 1C joined, and me drawn parallel to «C :Co is tiie cosine ofthe latitude to the radius CP, and CY that co
sine augmented in the ratio before named ; YQ being to Y/, thatis, Ce to Cn or CP, as the tangent of the angle YCQ, the latitudeof the point E to tiie tangent of the angle S’Ct belonging to theaugmented cosine. Thus, if M represent the measure in a latitudedenoted by E, and N tiie measure at the equator, let A denote anangle whose measure is
cos. Ex 3
c/
M
tan.
1 lien
A .
less axis
N tan. E greater axis
Rut M, or the length of a degree, obtained by actual mensura-tion in different latitudes, is known from tiie following table:
l
A'amts.
I Maupertuis, &c.
Cassini and ) (.
La Cable $ \ .
■ Roscovicli.
De la Cable.
■ Juan and Ulloa.
I CoiuUmiine.7.
1 All it.
Cu hi c of M.
t> t
toises
fit) 20
M = 57458
49 22
M = 57074
45 00
M = 57050
43 00
M = 56972
33 18
M = 571-37
( at the 4
M = 56768
5 I’qua- [•
M = 5G753
l tor. )
M = 5G749
Now, by comparing the 1st with each of the following ones; the2d with each of the following; and in like manner tiie 3d, 4th,and 5tli, with each of the following; there will he obtained 25 re-sults, eaeli shewing the relation of the axes or diameters; thearithmetical means of all of which will give that ratio as 1 to0 -pp;, 1989 . If the measures of tin- latitude of 49° 22', ami of 45",which fail within the meridian-line drawn llnimgh France , andwhich have been re-examined and corrected since the noitliernanil southern expedition, he compared w ith those of Maupertuisand his associates hi the noitii, and that of Bouguer at the equator,there will result six dillerent values of the ratio of tiie two axes;the arithmetical mean of -all winch is that of 1 to 0-9953467, whichmaybe considered as tiie ratio of the greater axis to the less;winch is as 230 to 228-92074, or 215 to 214, or very near the ratioas assigned by Newton. Now, the magnitude, as well as thefigure of the earth, that is, the polar and equatorial diameters,may lie deduced from the foregoing problems. For, as half (lieI at us rectum of the greater axis An is tiie radius of curvature at A,it is given in magnitude from the degree measured there, and
thence