406 REPORT—1883. 
method of treating it. This method was an extension of that made familiar to us 
in the writings of Sir John Lubbock and used by Dr. Whewell. Their system, 
however, was imperfect, inasmuch as it did not introduce what was most necessary, 
viz., a fourfold classification of the tides, as follows: 1, a lunar direct tide; 2, a 
lunar obverse tide ; 3, a solar direct tide; and 4, a solar/obverse tide. The author 
claimed to haye shown that this classification, inasmuch as it introduced the con- 
sideration of the difference between the lunar action, direct or obverse, when in the 
southern hemisphere, as contrasted with the corresponding action when in the 
northern hemisphere, gave a clue to the daw of the diurnal inequality, and when 
the effects were formulated in a series of tables the heights of successive tides 
could be predicted with unprecedented accuracy. The author explained how he 
had been helped in finding out this law of the inequality by means of a graphic 
process, which exhibited the varying positions of the sun and moon, drawn to 
scale, and the resulting heights of lunar and solar, direct and obverse, tides; and he 
disclaimed any opposition in his method of treating the subject to the more exten- 
sive theory favoured by the British Association; suggesting only that, as his (the 
author's) observations were confined to Liverpool and the west coast of Lancashire, 
an effort should be made to carry out a system of observations in other parts of 
Great Britain. He regretted that this effort should languish for want of funds, 
which ought to be forthcoming to the needful amount from the Association, since 
amateurs were left to do so much at their own private expense. 
The author then described his self-registering instrument, which had been 
kindly placed in a suitable position by the harbour authorities at Fleetwood, the 
whole constituting a suitable tidal observatory, and the registers were compared 
month by month with those obtained at George's Pier, Liverpool. The paper 
concluded by stating that the system of tables thus originated had been adopted 
by the authorities of the Hydrographic Department of the Admiralty for the last 
six years, for Liverpool only, and had been inserted in the annual volume of Tide 
Tables published by their authority, 
SATURDAY, SEPTEMBER 22. 
The following Report and Papers were read :— : 
1. Report of the Committee on Mathematical Tables—See Reports, p. 118. 
2. On Lamé’s Differential Equation. By Professor Linprmany.—See 
Reports, p. 351. 
3. On a Fundamental Theorem in the Dynamics of Non-Euclidian Space. 
By Professor Rozgrr 8. Batu, DD.D., F.R.S. 
The theorem contained in this paper has been familiar to the author for two or 
three years. He had always thought hitherto that it must have been known to 
mathematicians, as it seems to be of fundamental importance in elliptic space. It 
is true that he never could find any reference to it, but he had been disposed to 
attribute this to his ignorance of the literature of the subject. Professor Linde- 
mann, who has done so much for this theory, had, however, assured him that the 
theorem is new. 
The effect produced on a rigid system by a pair of equal rotations about two 
right lines which are conjugate polars to the absolute is called by Clifford a 
right vector. Ifthe rotations are equal, but with opposite signs, they constitute a 
left vector. A pair of equal forces on two conjugate polars we may call a right 
couple, or a left couple, according to the way in which the forces are directed. 
The theorem now submitted is thus stated: The virtual moment between any 
right vector and any left couple, or between any left vector and any right couple, 
is equal to zero. 
