16 Proceedings of the Royal Irish Academy. 



point dining its motion from one limit to the other, the time integral 

 of the energy diminished by the time integral of the component of the 

 velocity along the path of the representative point into the element of 

 that path (- Scrdp), depends on the mamier in which the point moves 

 in the given interval of time from one limit to the other ; bnt when 

 the fliiid motion is irrotational the difference of these integrals (49) 

 is independent of the mode of passage of the representative point. 



It should he observed that JScrdp is not now the j^ozi^, for the 

 velocities o- are taken successively in time and not at the same instant 

 along the path. 



"When we do not suppose the motion irrotational, the difference of 

 the integi-als (Ai't. 3.) for two different modes of passage is given by 



Q, = (2-T- [|s(S;;dp-d;;Sp)('|^+V^VjJSVV(7YdpSp. (55) 



In this case by (51) and (52) 



^ + V^=ycroyVc7, (56) 



and this relation may be employed to simplify (55). But the double 

 integral admits of further simplification, for if in the variation of the 

 mode of passage we suppose the curves 8p to be instantaneous or to 

 pass through the loci of representative points at eveiy instant during 

 the passage, we shall have ht - 0, so that 



Q, = Q- J-J df S SpYa.YVa- - jj S Y Vcr Y dp Sp . (57) 



Art. 12. — To illustrate the meaning of the double integral we 

 take the simple case (35) which becomes in tenns of t and p 



Q = J7 S 0-1 (d^ d'p - d't dp) + jj S (7, Y dp d'p (58) 



because 



(d^' d'q) = d^ d'p - d't dp , [d^ d'j] = Ydp d'p . (59) 



The limits being fixed must consist of a closed curve composed of 

 pairs of corresponding points of departure and arrival, the times being 

 prescribed for every point. There is now a singly infinite system of 

 representative points, each of which leaves its point of departure at a 

 definite • instant and reaches its point of arrival at another definite 

 instant, and when the mode of passage is given the path and the 

 rate cf description of that path is given for eveiy representative 



