1867.] On the Lunar Atniosplieric Tide at Melbourne. 



489 



We have above a very simple example of a general principle, viz. that in 

 order that a partial differential equation of the second order, or a pair of. 

 simultaneous partial differential equations of the first order, may admit of 

 a solution containing arbitrary functions, the coefficients must satisfy a cer- 

 tain equation of condition ; from which it follows that, except in the sim- 

 plest instances (in which the terms of the equation of condition vanish), 

 there is a moral certainty that such a differential equation or pair of equa- 

 tions which have not been specially selected for the purpose, and whose co- 

 efficients do not involve a disj)osable quantify by which the equation of con- 

 dition may be satisfied, will not admit of a solution involving arbitrary 

 functions. 



The equations applicable to the motion of an elastic fluid along the axis 

 of a tube afford a remarkable illustration of the scope of these remarks. 



Those equations consist of a pair of partial differential equations of the 

 first order involving Jive variables, viz. y, t, p, v, ]) ; and it may be shown 

 a priori, that when derived upon a true theory they must be capable of a 

 solution containing two arbitrary functions ; from which it follows that a 

 third equation will require to be satisfied. For this purpose we have p, the 

 pressure, ready to our hands. 



From the fact of the existence of the equation of condition not having 

 been suspected by the founders of the theory of fluid-motion, at the same 

 time that it was absolutely necessary for them to assign a form to p, they 

 had recourse for that purpose to an empirical method ; thus, on the one 

 hand, depriving us of the power of satisfying the requirements of the pro- 

 blem, and on the other, abandoning the means for the determination of p 

 which the analysis furnishes. 



It cannot be matter of surprise that the law of pressure suggested under 

 these circumstances should be entirely erroneous, as (by two other inde- 

 pendent methods, one founded upon purely physical, the other upon purely 

 analytical considerations) I have elsewhere shown. 



III. On the Lunar Atmospheric Tide at Melbourne.'' By Dr. G. 

 Neumayer^ late Director of the Flagstaff Observatory, Mem. 

 Acad. Leop. Communicated by Lieut. -Gen. Sabine, President. 

 Received April 10, 1867. 



Anxious to assist the development of so interesting a branch of know- 

 ledge on tlie connexion of forces in nature as the influence our satellite 

 exerts upon the earth's atmosphere, I had made it a point to include 

 investigations, tending to facilitate studies in this direction, in the plan of 

 discussion of the observations made at the Flagstaff Observatory about to 

 be published. Fully aware that a geographical position, such as that of Mel- 

 bourne (37° 48' 45" south lat. and D" 39'" 53' east long.), affords but very 

 few chances for arriving forthwith at a result which might be regarded as 

 final, I thought it nevertheless of the highest importance to decide how 



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