564 REPORT — 1888. 



so great that it may be treated as infinite, the motion is represented by a harmonic 

 function whose amplitude diminishes as the time increases. Professor Stokes' result 

 ■was obtained by means of the dissipation function ; and the object of the present 

 paper is to solve the problem by a direct process, which possesses the advantage 

 of furnishing a result which is applicable to highly viscous as well as to shghtly 

 viscous liquids, and also to solve the same problem when the depth of the liquid is 

 finite. 



In order to solve the problem of deep sea waves we must find a solution, repre- 

 senting wave motion, of the equation 



v2(v=-i'-'rf/(/0V' = (1> 



■where y(r is Earnshaw's current function, and which also satisfies the conditions 

 that the tangential and normal stresses at the free surface are zero. Measuring 

 the axis of .r in the direction of propagation of the waves, we assume that .r and t 

 enter in the form of the exponential factor fn^'+'-i, and the principal object of the 

 investigation is to find the equation for determining k. This equation is 



/f^ + 4OT-Av+irm + 4wiV-4HtVa = . . . . (2> 



where a- = m'^ + klv (3) 



When V is small, as in the case of water, 



k= -2tn"v±^(4m^v"-g7ti) . . . . (4> 

 = —2)n"v±in (5) 



approximately, where n' = (/)n ; which represents a train of waves whose velocity of 

 propagation is slightly less" than (^X/27r)*, and whose modulus of decay is X'/Stt'j', 

 where X is the wave length. 



If, however, v is large, we shall obtain 



A>=-Mj'x-92 (G> 



approximately, which shows that the value of k is real and negative ; hence the 

 motion is non-periodic, and rapidly subsides. 



When the depth of the liquid is finite and equal to h, the equation for Jc is a 

 rather complicated transcendental one, which is expressible in the form of a 

 determinant of four rows. If, however, v is small, it is reducible to a quadratic 

 whose roots are 



k= — ^Di^v + i{mg tanh mhy> .... (7) 



•which is the approximate solution of the problem. 



This represents a train of waves whose velocity of propagation is 



(^X/27r . tanh 2nhlK)\ 



and whose modulus of decay is X^/Stt";'. 



6. On a Hydrostatic Balance} By J. Jolt, M.A., B.E. 



7. On the Meldometer. By J. Jolt, M.A., B.E. 



This is an apparatus enabling mineralogists to compare the melting-points of 

 minerals and observe their behaviour at very high temperatures. It is applicable 

 for dealing with very small quantities of bodies. The apparatus consists essentially 

 of a ribbon of thin platinum, about 4 cm. long by 3 mm. wide, clamped in two 

 forceps with attached binding screws, so that a current from a small storage cell, 

 or two or three Grove's cells, may be passed through it. The apparatus rests on 

 the stage of the microscope, the fragments of mineral being laid on the platinum 

 strip and observed with a one-inch objective. A resistance, of simple construction, 

 which can be set to automatically diminish or increase at any desired rate is placed 



' An account of this apparatus appears in the Philosojildcal Magazine for 

 September 1888, p. 26. 



