110 On the Distribution of Energy in a mass of Liquid. 



ov 

 drawn through two consecutive moving particles ; and -p^ is 



the angular velocity reckoned in the same direction of a line 

 drawn perpendicular to the tangent through P and Q, two con- 

 secutive moving particles. The sum of the two is therefore 

 twice the molecular rotation ; and if we call the molecular ro- 

 tation {2j , we shall have 



M = 2.1l^> (1) 



9 



Now v . PQ is constant, being the flow in an elementary stream 

 of breadth unity; and hence we see that the difference of energy 

 between two consecutive elementary streams is proportional to the 

 molecular rotation at any point of either. 



An immediate consequence of this is that the molecular ro- 

 tation is the same at all points situated in a line of motion, and 

 can be determined when the difference of head is known. Thus 

 in the well-known trochoidal motion, the difference of head 

 between one trochoidal layer and the next consecutive is easily 



2r 

 seen to be p- . 8r, where r is the length of the tracing-arm, 



and R the radius of the rolling circle. Hence, placing this in 

 equation (1), the known value of the molecular rotation is 

 more easily determined than in any other way. 



Again, we learn that if the energy of any portion of the 

 liquid be initially uniformly distributed, the motion of that 

 portion must be irrotational and must always remain so ; and 

 thus we have a simple demonstration that the permanent mo- 

 tion of a perfect liquid past a solid free from discontinuity of 

 curvature and perfectly smooth is necessarily irrotational. This 

 demonstration clearly also applies to any case in which a per- 

 fectly smooth and fair-formed solid moves with a uniform mo- 

 tion of translation through a liquid at rest. 



The peculiar form in which the condition for irrotational 

 motion is expressed, namely 



leads to a conclusion of much importance — namely, that by a 

 surface of " fair " form is to be understood a surface free from 

 discontinuity of curvature, and not merely from discontinuity 

 of form : thus a circular arc joined to a straight line is not a 

 fair form, since the radius of curvature changes abruptly and 

 occasions a discontinuity of form in the next consecutive 



