AMERICAN ASSOCIATION TOE THE ADVANCEMENT OE SCIENCE. 71 



unsatisfactory. Nor were these discrepancies without law, as representing their 

 residuals by curves did not fail to show. By introducing corrections for declination 

 and parallax of the moon increasing and decreasing, we reduced these discrepancies, 

 but still the results were not sufficient approximations. With the numerical reduc- 

 tions of the observations before referred to, was commenced in 1853, under my 

 immediate direction, by Mr. L. W. Meech, a study of the theory of the tides, di- 

 rected chieflv to the works of Bernoulli, La Place, Avery, Lubbock and Whewell. 

 The immediate object which I had in view was the application of the wave theory 

 to the discussion of our observations. I thought that the mind of an expert mathe- 

 matician, directed entirely to the theoretical portions of this work, with direction- 

 by a physicist, and full opportunities of verifying results by extended series of ob- 

 servations, the computations of which should be placed by others in any desired 

 form, would give, probably, the best result in this combined physical and mathe- 

 matical investigation. 



The general form of the different functions expressing the tidal inequalities is the 

 same in the different theories, and may be said on the average to be satisfactory as 

 to the laws of change which these inequalities present. Whether we adopt, with 

 La Place, the idea that periodical forces produce periodical effects, or with Avery, 

 that the tidal wave arrives by two or more canals ; or with Bernoulli and Lubbock, 

 the results of an equilibrium spheroid; or with Whewell, make a scries of inequali- 

 ties, semi-menstrual, parallax and declination, with different epochs, we arrive at the 

 same general results, that the heights and times of high water may be represented 

 by certain functions, with indeterminate co-efficients, in the form of which the 

 theories in a general way agree. By forming equations from the observations, and 

 obtaining the numerical values of the co-efficients by the methods used so common- 

 ly in astronomical computations, the result is accomplished. 



A general consideration of the co-ordinates in space of the moon and sun, with- 

 out any special theory, would lead to the same result, representing the luni-tidal in- 

 terval by a series of sines and co-sines, with indeterminate co-efficients. 



The grouping of the observations of one year at Boston, to apply this method- 

 the formation of the equations and their solution by the method of indirect elimi- 

 nation has been the work of Mr. R. L. Avery. 



To test the co-efficients, computations, for the predicted times of the tide at Boston 

 harbor were made for a period from March 1853, to January 1854, and from com- 

 parison of these with the observed, it appears that in twenty pairs of tides, the 

 morning and afternoon being grouped to get rid of the diurnal inequality, there are 

 two differences of less than two 2 m., thirteen of more than 2 m. and less than 4 m., 

 three of more than 4 m. and less than 10 m., two of more than 10 m. The proba- 

 ble error of the prediction of a single pair of tides is 4.12 m. so that greater accuracy 

 of prediction has been attained by this method from a single years's observations that 

 was found at London bridge from a period of nineteen years. 



LAW OF MOUTALTTY. 



Prof. McCoy, of Albany, read a paper in which he announced the important 

 discovery of a mathematical formula which correctly expressed the law of mor- 

 tality for all ages ; it was first evolved from an analysis of the Carlisle and 

 Northampton tables, but the Professor had compared it with a large number of 

 others and said that, "so complete is its agreement with all, that at no age does 

 the calculated number of the living differ from the number given iD the tables by 



