In order to obtain the term in R depending upon 4, 72, 7,2 and 74, it is sufficient to 8 + ri cos &+77, (22 + y2) cos (n − n,) cos (r− 2 n + 2 n) +, &c. } { 2 b5,0 + b5,1 cos & + b5,2 cos 2 r +, &c. } 5 y2 y2 + 2 i — Y Y 1 (y2 + y2) cos (r + n − n) — 271 (y2+y,2) cos (r− n+n,) 4 4 3 +2n)} + (2 +22x2 + 72) cos r + 2 y2 y cos (2 n − 2 n) + cos (2 r — 2n+2n) } 4 a2 3 a2 3 a3 a3 15a2 b5,0 за 9 a2 } y2 y2 64 as b5,0 + 3 a2 32 a3 b5,0 3 a2 64 a3 9 a2 64 a3 3 a2 b5,2 64 as b5,2-64 a3b5,0 32 a3 b5,2 + 3 a3 64 a3 3 a2 64 a3 3 a3 b5,2 32 ap 45,2 } 77, (72 + 2 3 a9 γγι 3 a2 a2 +{ 3 a2 3 a3 32 a 3 a3 a2 16 a$ 65,0 +32 a$ 65,2 7, +16 a3 65,087, (22 + y,2) cos (7 − n +n,) 32 a3 65,0 y2 y2 cos (2 r — 2n+2n). In order to give another example of the employment of this method, I propose to calculate the coefficient of the argument of which occurs in Professor AIRY's inequality of Venus, nt and n ̧t being the mean motions of that planet and of the earth. It is easily seen from the preceding pages that R contains the term and substituting my notation in Professor AIRY's expression, that which I have found results. The method I have given of developing the disturbing function in terms of the mean longitudes may also be employed with advantage in procuring the development in terms of the true longitudes. In this problem 3 e2 (2入 2 ) + cos (4λ — 4 ₪) } = log. a - - 13e (1 + 2) cos (2 − ) + — (1+) cos (2-2) es 4 32 -e -cos (3-3)+cos (4 2. — 4 m) Se €2 4 dr = - 3/4 - 13e − e ( 1 +2) cos (2) + (1 + e2) cos (2 λ — 2 ) rde 2 It follows from the analysis of M. POISSON, in his Mémoire sur le Mouvement de la Lune autour de la Terre, that the coefficient of cos (2-2) in the development of the quantity according to the true longitudes, is the same as that of cos (2-2) in the development of R according to the mean longitudes. By means of these equations, and after reductions similar to those of which so many examples have been given in the course of this paper, I find the coefficient of e2e2cos (22-22,-2λ+2+2λ,-2) or e2e,2cos (2-2) 3 a3 16 9a4 64 a 2 b5.2. 9a4 32 « b52 V. On the Results of Tide Observations made in June 1834 at the Coast Guard Stations in Great Britain and Ireland. By the Rev. WILLIAM WHEWELL, F.R.S., Fellow of Trinity College, Cambridge. Received March 27,-Read April 2, 1835. IN the conclusion of "An Essay towards a first Approximation to a Map of Cotidal Lines," published in the Philosophical Transactions for 1833, I stated my opinion that simultaneous tide observations, made at the stations of the Preventive Service, and continued for a fortnight, would give us a clearer view of the progress of the tide along the coasts of this country than we could acquire from any records then extant. A representation to this effect being made to Captain BowLES, the Chief Commissioner of that Service, and to Captain BEAUFORT, the Hydrographer of the Admiralty, those gentlemen entered with great interest and activity into the proposal for promoting this branch of science by such a series of observations; and they undertook to give orders for carrying the plan into effect, and directions for its execution. Such observations were accordingly made at all the Preventive Service stations on the coasts of England, Ireland, and Scotland, from June 7 to June 22 inclusive, and the registers of the observations were sent to the Admiralty, where they now are. I expected to be able to deduce from these returns the solution of several curious and important questions respecting the tides, and probably to obtain some new laws of their phenomena. For this purpose, however, it was necessary to perform a previous reduction of the registered observations, correcting the times as far as the methods employed would allow, and subtracting from each time of tide the time of the previous transit of the moon, in order to obtain the interval. Though this operation was very simple, the performance of it in so many cases (above 12,000) required more time than I could devote to it. Captain BEAUFORT kindly allowed it to be executed by Mr. DESSIOU, of the Hydrographer's Office; and it was my intention to defer laying the account of the observations before the Society till the whole of them had been reduced, and their results investigated. But Mr. DESSIOU, having executed this reduction for the whole of the south coast of England, has been prevented by illness and by more pressing employments, from proceeding to the remaining coasts. In the mean time, having examined the reduced observations, I have been led to some conclusions which appear to me interesting and important; and which, I think, considering the delay which may attend the reduction of the remaining returns, and the intention which is entertained of repeating the observations in the ensuing June, it may be worth while very briefly to announce. I shall defer the communication of |