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LXXIV. On sojne Points relating to the Theory of Fluid 

 MotioTi. % the Rev. J. Challis, M.A., F.R.S., F.R.A.S., 

 Plumian Professor of Astronomy and Experimental Philo- 

 sophy in the University of Cambridge^. 



IN a memoir on certain questions in the theory of the motion 

 of fluids, published (1847) by Professor P. Tardy of Flo- 

 rence, for a copy of which I am indebted to the author, re- 

 ference is made to a communication contained in torn, xxiii. of 

 the Comptes Rendus of the Academy of Sciences of Paris, in 

 which several of my investigations on fluid motion are brought 

 under review by M. J. Bertrand. Professor Tardy states at 

 the same time, that he had himself previously made remarks 

 on one of the points on which I am supposed to be in error. 

 The citation of Professor Tardy first made me aware that 

 M. Bertrand had taken notice of my labours. On turning to 

 the article in the Comptes Rendus, I perceived that the im- 

 portant errors [erreiirs graves) attributed to me were partly 

 due to misconception of my reasoning, which, I am ready to 

 admit, may not have been developed with sufficient clearness; 

 and partly to the circumstance, not unusual in the history of 

 science, that new truths appear to be errors so long as the 

 errors they replace are supposed to be truths. It will suffice 

 for the present to advert to one point of primary importance. 

 I have repeatedly contended that, to complete the analytical 

 theory of hydrodynamics, a new general equation is absolutely 

 required, M. Bertrand first calls in question the principles 

 on which this equation is established, and then contends that 

 an equation which I derived from a combination of the new 

 equation with that which is usually called the equation of con- 

 tinuity, is identical with a particular case of this latter equa- 

 tion. The following investigations will supply answers to 

 these objections. 



1. I propose first to exhibit the principles on which the 

 new equation rests, and to deduce it accordingly. If the equa- 

 tion i>{x,y, 2f t) = express any given relation between the 

 coordinates a:, y, z and the time t, any other relation be- 

 tween the same quantities may be expressed by the equation 

 ^{x + da;, y + ^y, z + dz, t + df) = 0, the increments dx, 8j/, dz 

 being in general functions of the coordinates and the time. 

 Supposing the increments to be indefinitely small, we obtain 



* Communicated by the Author. 



