Haughton — On Complete Tidal Equations. 231 



XXXVIII. — Ajt Easy Mode of obtaining the complete Dipfeeential 

 Equations of Motion of an Ocean sueeounding a Solid JSTijcleus, 

 and subject to ant Distuebing Foeces (the ISTtjcleus itself 



EEVOLVING ON A ElXED Axis); WITHOUT CALCULATION OE TeANS- 



foemation of Co-oedinates ; feom simple Geometeical and 

 Mechanical Peinciples. By the Eev. Samuel Haughton, 

 M.D., Dubl. ; D.C.L., Oxon. 



[Eead April 14, 1879.] 



The complete differential equations of the motion of the sea or 

 atmosphere, referred to polar co-ordinates, are regarded, justly, as one 

 of the most brilliant results that we owe to the genius of Laplace ; 

 and yet they are found to be a "stumbling-block" in the way of 

 young mathematicians, from the hideously repulsive form in which 

 they are deduced, by transformation, from fixed rectilinear co- 

 ordinates, by Laplace himself, and by his followers. 



Any attempt, therefore, to write down these equations at sight, 

 from elementary geometrical and mechanical principles, will be re- 

 garded as useful. 



According to the self-evident principle of D'Alembert, all prob- 

 lems of Dynamics are reducible to problems of Statics, by intro- 

 ducing velocities and accelerating forces, equal and opposite to the 

 existing velocities and accelerating forces. 



Now, the most general equations of equilibrium, of any system, 

 are the following, six in number — 



X = 0, r = 0, Z = Q, 



Z = 0, Jf= 0, i\r= 0, (1) 



where X, Y, Z, are the sums of the external forces resolved along 

 three rectangular axes ; and Z, M, N, are the sums of the Couples (or 

 Twists) of those forces round the axes of X, Y, Z, respectively. 

 The corresponding dynamical equations are — 



_ d ( (ly (h\ 



^ -Tt ['di-'^dtj = '' 



,_ d I dz dx\ 



dt \ dt dt I 



,^ d I dx dy\ 



