Prof. Challis 6)i the Principles of Hydrodynamics. 233 



This result shows that the vakie of s does not change in pass- 

 ing from one j^oint to another of a given surface of displacement 

 at a given time. Hence if A and B be any two points, either at 

 a finite, or indefinitely small distance from each other, on one of 

 a series of surfaces of displacement of which the lines s are ortho- 

 gonal trajectoiies, then the trajectories terminating at A and B 

 ai-e equal to each other. So also the parts of the same two tra- 

 jectories terminating at the points a and b of another surface of 

 displacement ai'e equal to each other. Consequently the parts 

 Aff and B6 between the two surfaces are equal. Now this cannot 

 be the case for any two points A and B unless the sui-faces be 

 parallel and the trajectories be rectilinear. Consequently the 

 lines s dra^m in the direction of the motion are at each instant 

 rectilinear. 



We have thus arrived at a particular circumstance of the 

 motion by assuming mlv + vdy + ivdz to be integrable indepen- 

 dently of the nature of the disturbance. But the circumstances 

 of the motion, which are independent of the disturbance, relate 

 to the mode of action of the parts of the fluid on each other. 

 We may therefore conclude, that the parts of the fluid act upon 

 each other in such a manner that the orthogonal trajectories of 

 the sui-faces of displacement are either instantaneously or per- 

 manently straight lines, if that action be such as to satisfy the 

 condition of making udx -\- vdy -\- ivdz integrable. But we have no 

 right to say that the mutual action of the parts of the fluid on 

 each other always satisfies that condition. 



To carry on the investigation of the law of the motion, so far 

 as the motion is independent of the nature of the disturbance, it 

 will next be required to solve the following problem. 



Proposition VIII. To obtain a general equation which shall 

 express at once that the motion is consistent with the principle 

 of constancy of mass, and that the directions of motion are nor- 

 mals to continuous surfaces. 



This may be done in two ways ; either by independent ele- 

 mentary considerations, or by means of the formulse already ob- 

 tained, which, being perfectly general, ought to suffice for this 

 pui-pose. For the sake of illustration I shall first make use of 

 the former method. 



Conceive two surfaces of displacement to be drawn at a given 

 instant indefinitely near each other, and so that one of them 

 passes through a point P given in position. On this surface 

 describe an indefinitely small rectangular area having P at its 

 centre, and having its sides in planes of greatest and least cur- 

 vatui'e. Draw normals to the surface at the angular points of 

 the area, and produce them to meet the other surface of displace- 

 ment. By the property of continuous cui-ve surfaces these nor- 



