"This table was computed from the following formula for the perturbations of the mean longitude and radius vector of Uranus, which are arranged in a form similar to that proposed by Leverrier, and adopted in his theory of Mercury. The mean longitude of each planet is denoted by the appropriate symbol of the planet. The elements of Neptune which are adopted are those last given by Mr. Walker, and the mass of Neptune which is introduced into the formulæ is zooooth of the sun's mass, for which any other mass is readily substituted by simple multiplication. "The perturbation of the mean longitude = 8v= + + 0.02 cos. (H) -818.98 sin. 2 (HK) -0.99 cos. 2 (-) + + + (H 0.00434 t sin. (—)-0.035411 t cos. (1⁄2 — ₪) +k sin. (+0—1) +k, sin. (2 + 0, 2TH) +k2 sin. (+2 — ̃œ) in which +14976 cos. (2 - H A=2692.74 sin. (2--) +149.76 c H E) 3 TH +106.80 sin. (4 K-2H-2)-43.08 cos. (4-2-2 σμ) 6.09 sin. (6-3-3)+ 4.20 cos. (6—3 H H 0.48 sin. (84-4)+ 0.47 cos. (8-4— 4 ₪μ) 0.06 sin. (10 -5 -5)+ 0.01 cos. (10-5-5☎μ) + k sin. 02.58 sin. (HK) 0.44 cos. (HK) -11.35 sin. 2 (HK) 0.02 cos. 2 (HK) 18.27 sin. 3 (HK) + 3.68 cos. 3 (HK) 0.78 sin. 6 (-) 0.56 cos. 5 (-) 0.17 cos. 6 (-) 0.10 sin. 8(-) 0.03 cos. 8 (μ— Ķ) 0.04 sin. 9 (-) 0.02 sin. 10 (-) 0.02 sin. 11 (— Ķ) - 0.01 sin. 12 (μ —Ķ) — k cos. — 0.42 sin. +0.02 sin. 3.68 sin. +13.42 sin. (H +1.44 cos. (K) 2 (μ — Ķ) — 11.35 cos. 2(-) 0.56 sin. 5(-)+ 284 cos. 0.17 sin. - 3(K) 4 (-) 5 (HK) 6 (-) 0.82 cos. 0.01 sin. 10 (-) 0.01 sin. 11 (HK) 0.04 cos. 10 (-) 0.02 cos. 11 (HK) 0.01 cos. 13 ( 0.65 cos. -1.07 sin. 5(-)- 0.48 cos. H -5.81 sin. 6 (-)- 2.50 cos. -K) -0.10 sin. 7 (-) 0.06 cos. 7 (HK) -0.06 sin. 8(-)-0.03 cos. 8(-) -0.03 sin. 9 (-) 0.02 cos. 9 (-) 0.01 sin. 10 (-)-0.01 cos. 10 (-) -0.01 sin. 11 (K) 0.65 sin. 2 (-)-0.86 cos. 2 (-) -0.13 sin. 3 (-)+0.49 cos. 3(K) -0.48 sin. 5(-) + 1.07 cos. 5 (-) -2.50 sin. 6 (-)+5.81 cos. 6 (-Y) -0.06 sin. 7 (H− ) + 0.10 cos. -0.03 sin. 8(-)-0.06 cos. -0.02 sin. 9 (-) + 0.03 cos. 9 (-) -0.01 sin. 10 (HK) + 0.01 cos. 10 (-Ę) +0.01 cos. 11 (μ—K) · ·7 (H—K) 8(-) k2 sin. 0,= +0.71 sin. (4-2-2)-0.01 cos. (4-2 # — 2 –μ) +0.38 sin. (84-4)+0.54 cos. (8-4-4≈ƒ) -0.03 sin. (10-5-5)-0.06 cos. (10-5-ɔ̃☎μ) 2H 13 k2 cos. 6. = +0.01 sin. (4 Ķ—2—2)+0.71 cos. (4 1⁄2 −2 −2′′μ) -0.54 sin. (84-4)+0.38 cos. (8-4 — 4 ̃μ) +0.06 sin. (10-5 -5 ☎ πμ ) — 0.03 cos. (10 к — ɔ̃ ƒ — 5 ̄μ) +0.031823 cos. 2 (-)- 0.000036 sin. 2 (- Ẹ) -0.000089 sin. (4 Ę - 2 w +0.001684 cos. (2) ¥ - 2 μ) — 0.000151 cos. (4 — 2 μ —2 ≈μ) × +0.000012 sin. (6 −3 −3 ) +0.000013 cos. (6 K-3-3) -0.000002 sin. (8 K-4 -4)- 0.000003 cos. (8—4—4) H H k' cos. +0.000009 sin. (-Ę) ' = H -0.000039 cos. (Ḥ — Ę) +0.000096 sin. 3 (-)-0000477 cos. 3 (-) -0.000001 sin. 4 (-)-0.000012 cos. 4 ( — Ę) -0.000033 sin. 5(-)+0.000158 cos. 5 (-) -0.000014 sin. 6 (μ —§)+0.000047 cos. 6 (- Ķ) 0.000004 sin. 7 ( ƒ — Ę) +0.000018 cos. 7 (-Ę) – 0.000002 sin. 8 (−)+0.000009 cos. 8 (-Ķ) -0.000001 sin. 9 (-)+0.000003 cos. 9 (μ — K) · 0.000001 sin. 10 ( ƒ — Ķ) † 0.000002 cos. 10 ( μ — Ķ) +0.000001 cos. 11 (HE)+0.000001 cos. 12 ( — K) - -0.000011 cos. ( H · -K) k' sin. ' -0.000219 sin. (HK) — K ) -0.000399 sin. 3 (-)- 0.000096 cos. 3 ( K ) k', cos. ',— — 0.000002 sin. 3 (-)- 0.000001 cos. 3 (—) -0.000009 sin. 5 (μ — И)+0.000027 cos. 5 (F — K ) -0.000005 sin. 7 (-)+0.000012 cos. 7 ( -0.000002 sin. 9 (-)+0.000006 cos. 9 ( -0.000001 sin. 10 (ы — )+0.000004 cos. 10 ( — K) +0.000003 cos. 11 (—) -K) +0.000001 sin. (2 K —н—≈н) +0.001854 cos. (2-ƒ—ãƒ) -0.000009 cos.(4 K-2 н-2₪ -0.000004 sin. (8-4-4)+0.000009 cos. (8-4—4₪μ) k', sin. 0, +0.000001 sin. 3 (-)- 0.000002 cos. 3 (HK) = +0.000027 sin. 5 (μ — K) +0.000009 cos. 5 (- Ķ) - +0.000012 sin. 7 (μ — Ķ) +0.000005 cos. 7(K) +0.000004 sin.10 (-)+0.000001 cos. 10 ( μ — И) +0.001324 sin. (2-)+0.000027 cos. (2 ——ãƒ) w k'2 cos. '2+0.000253 sin. (2--~μ) -0.000660 sin.(4—2—2) −0.002992 cos. (4-2-2μ) +0.000105 sin. (6K-3-3)+0.000282 cos. (6 к—3н—3₪μ) -0.000017 sin. (8—4—4) -0.000025 cos. (8-4-4≈μ) +0.000002 sin.(10-5-5)+0.000003 cos. (10-5н–5≈μ) k', sin.' -0.005710 sin. (2--) +0.000253 cos. (2—н—≈н) -0.002520 sin.(4-2-2)+0.000558 cos. (4-2-2) +0.000254 sin. (6-3-3) -0.000085 cos. (6 K-3-3₪μ) -0.000025 sin. (8-4-4)+0.000017 cos.(8-4μ–4≈μ) +0.000003 sin. (10-5-5)-0.000002 cos. (10-5-5σμ) 43 Mr. Pierce remarked that his original views were unchanged in regard to the importance to be attached to the vast discrepancies between the predicted and observed orbits of the planet which disturbs the motions of Uranus. "Neptune is not the planet designated by geometry, although it is a perfect solution of the problem which analysis had undertaken to investigate, and had really solved, but in a form radically different from the actual solution of nature. This is not a personal question; it is certainly not one in which the reputations of Adams and Leverrier are concerned. The accuracy of their investigations is not assailed; but it is expressly admitted that they announced the correct results of most profound analytical researches. "The fair consideration of this question cannot be made without recalling the true office and position of geometry in science, which alone entitles it to the appellation of the key to the physical world. Mathematics is the science of exact measurement; accuracy is its sole aim and object, and it is this which places it in harmony with a creation, which is subject to perfect law and undeviating order. An inaccurate result cannot be a geometrical one; a result, inaccurate beyond certain well-defined limits, does not belong to the exact science; an inconsistency, which exceeds a certain amount, may not be neglected by him who deals with nothing but more or less, without disturbing the very foundations of his faith. "The geometrical statement was distinctly made, that the planet which disturbed Uranus could not be at a less mean distance from the sun than thirty-five times the earth's mean distance from the sun; that is, that no planet which was within this distance could cause the observed irregularities in the motions of Uranus. Neptune's mean distance from the sun is only thirty times the earth's mean distance, and yet Neptune does account for the perturbations of Uranus. It is five hun. dred millions of miles nearer the sun than it was distinctly stated by geometry that it possibly could be, in order to be capable of producing the effect which it actually does produce. The spirit of mathematical accuracy cannot be supposed to be sufficiently elastic to embrace so great an inconsistency, amounting to one sixth part of Neptune's distance from the sun, and to one half of the distance of his orbit from that of Uranus. "Whence comes this enormous difference between the theoretical |