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The earth's ellipticity
Oct., 1884.]
The Earth's Ellipticity.
"THE
EAI~TH~S
295
ELLIPTICITYY
By PLINY EARLE CHASE, LL.D. In the J u l y number of the JOU~NAI~ OF T~E F~A~KLIN INSTITUTE, Prof. d'Auria' has amended his first paper on the Ellipticity of planets, by introducing some of the omitted-elements to which I called his attention,* and thus obtaining Newton's value for e. Through the hypothesis that polar and equatorial gravitation were equal when the earth's ellipticity was determined, a hypothesis which seems to me altogether untenable, he gets a value nearly identical with Listing's. I n his letter to the Editor (p. 152) he shows, by virtually charging me with begging the question, that he totally misunderstands my reasoning. I do not "accept Neweomb~s [Listing's] estimate of the ratio of centrifugal force to attraction at the equator;" but I show that the probability of that estimate is confirmed by the closeness of its accordance with a result which is indicated by known equilibrating tendencies. The centrithgal and centripetal relations which he represents by j and k, being constantly operative~ the calculation may be simplified by considering their joint efficiency, instead of treating them separately and modi~ing one of them hypothetically, so as to bring about an accordance between theoretical and observed values. Let wo = angular velocity of free revolution at earth's equatorial surface, or self-sustaining, velocity under combined centrifugal and centripetal tendencies~ w being angular veloclty of terrestrial rotation. Then
combined tendency of 2" and k to compress the terreso
Wo
trial spheroid.
I f the earth were at rest we should have (__w~ 2 ---- e ~ 0 \ Wo/
and the form would be spherical ; if w ~---%, we should have e ~ 1 and the form would be that ot a flat disk. The actual value is [ w ~ ~ __
~Wo/
1 288"4.
The most recent estimates of 1- are the following: e
Fischer ................................. 1868 Listing ................................. 1872 Jordan ................................. 1878 Clarke ................................ 1880
Ante, p. 20.
288"5 288"5 286"5 293"5
296
Lengths of Indicator C a r d s .
[J'our.Frank.Inst.,
I can think of no principle which is more likely to produce and maintain such closeness of accordance than that of "the flow of solids." In each oscillation of semi-gravitating rotation the stun of the gravitating accelerations, gt, represents a centrifugal tendency (16' 987r)2 times as great as that of mere rotation. Prof. d'Auria may, perhaps, find in this fact and in the immense energy of mthereal oscillation a fruitful fie]d for the exercise of his skill in mathematical analysis. The harmonic connection of Neptune's mass with that of the earth~ seems to be dependent upon the fact that ~ethereal waves, as well as all other waves, tend to maintain the velocity, 1 / ~ , which is due to the place at which they originate. The value for the reciprocal of em~h's mass, 327994, which was given in the May number of the JotrRs*AG was based upon the hypothesis that the earth's orbit was circular; the modified value, 329196, made allowance for the secular orbital eccentricity.
L E N G T H S O F I N D I C A T O R CARDS.
By ROBERT G:RI2~ISYIAW, For eonvenienee in reference, I have arranged a table showing at a glance the length of diagram that will be given where the point of attachment of the cord to the main motion lever is ½, ½, ~, etc., as far from the pivot as the distance from the pivot to the point at which the vibrating link is attached to the main lever. Thus : For indicating an engine of 20-inch stroke, if the "pendulure" is 43 inches long between stationary and swinging eentres, if you attach the card { of the distance of 43 inches fi'om the fixed een~re, you get a card 3"33 inches long. I f you attach it { the distance, or 8"6 inches, you get a card 4 inches long; and if you attach it ¼ the distance, or 10~ inches, yqu get a closed tracing 5 inches long. One-fifth the distance would be most convenient on a Thompson or a Tabor machine, or any other instrument having a big barrel, if the speed is not too great, and ~ on a "Crosby No. 2." It is just as well to have the lengths known beyond any possibility of error in the hurry of calculation ; as it is mortifying to find a card
The Earth's Ellipticity.
"THE
EAI~TH~S
295
ELLIPTICITYY
By PLINY EARLE CHASE, LL.D. In the J u l y number of the JOU~NAI~ OF T~E F~A~KLIN INSTITUTE, Prof. d'Auria' has amended his first paper on the Ellipticity of planets, by introducing some of the omitted-elements to which I called his attention,* and thus obtaining Newton's value for e. Through the hypothesis that polar and equatorial gravitation were equal when the earth's ellipticity was determined, a hypothesis which seems to me altogether untenable, he gets a value nearly identical with Listing's. I n his letter to the Editor (p. 152) he shows, by virtually charging me with begging the question, that he totally misunderstands my reasoning. I do not "accept Neweomb~s [Listing's] estimate of the ratio of centrifugal force to attraction at the equator;" but I show that the probability of that estimate is confirmed by the closeness of its accordance with a result which is indicated by known equilibrating tendencies. The centrithgal and centripetal relations which he represents by j and k, being constantly operative~ the calculation may be simplified by considering their joint efficiency, instead of treating them separately and modi~ing one of them hypothetically, so as to bring about an accordance between theoretical and observed values. Let wo = angular velocity of free revolution at earth's equatorial surface, or self-sustaining, velocity under combined centrifugal and centripetal tendencies~ w being angular veloclty of terrestrial rotation. Then
combined tendency of 2" and k to compress the terreso
Wo
trial spheroid.
I f the earth were at rest we should have (__w~ 2 ---- e ~ 0 \ Wo/
and the form would be spherical ; if w ~---%, we should have e ~ 1 and the form would be that ot a flat disk. The actual value is [ w ~ ~ __
~Wo/
1 288"4.
The most recent estimates of 1- are the following: e
Fischer ................................. 1868 Listing ................................. 1872 Jordan ................................. 1878 Clarke ................................ 1880
Ante, p. 20.
288"5 288"5 286"5 293"5
296
Lengths of Indicator C a r d s .
[J'our.Frank.Inst.,
I can think of no principle which is more likely to produce and maintain such closeness of accordance than that of "the flow of solids." In each oscillation of semi-gravitating rotation the stun of the gravitating accelerations, gt, represents a centrifugal tendency (16' 987r)2 times as great as that of mere rotation. Prof. d'Auria may, perhaps, find in this fact and in the immense energy of mthereal oscillation a fruitful fie]d for the exercise of his skill in mathematical analysis. The harmonic connection of Neptune's mass with that of the earth~ seems to be dependent upon the fact that ~ethereal waves, as well as all other waves, tend to maintain the velocity, 1 / ~ , which is due to the place at which they originate. The value for the reciprocal of em~h's mass, 327994, which was given in the May number of the JotrRs*AG was based upon the hypothesis that the earth's orbit was circular; the modified value, 329196, made allowance for the secular orbital eccentricity.
L E N G T H S O F I N D I C A T O R CARDS.
By ROBERT G:RI2~ISYIAW, For eonvenienee in reference, I have arranged a table showing at a glance the length of diagram that will be given where the point of attachment of the cord to the main motion lever is ½, ½, ~, etc., as far from the pivot as the distance from the pivot to the point at which the vibrating link is attached to the main lever. Thus : For indicating an engine of 20-inch stroke, if the "pendulure" is 43 inches long between stationary and swinging eentres, if you attach the card { of the distance of 43 inches fi'om the fixed een~re, you get a card 3"33 inches long. I f you attach it { the distance, or 8"6 inches, you get a card 4 inches long; and if you attach it ¼ the distance, or 10~ inches, yqu get a closed tracing 5 inches long. One-fifth the distance would be most convenient on a Thompson or a Tabor machine, or any other instrument having a big barrel, if the speed is not too great, and ~ on a "Crosby No. 2." It is just as well to have the lengths known beyond any possibility of error in the hurry of calculation ; as it is mortifying to find a card