10 Proceedings of the Royal Irish Academy. 



{dq, Sq) and [dq, 8q'] by y8 - a and Yay8 respectively. (Compare 

 Trans. H.LA., xxxii., p. 5.) It is easy to see that tlie terms in 

 /8 - a and Ya/? in (14) must vanish separately, so we may replace this 

 xjondition by the pair 



SD.,i^(^,7)-SyD.i^(^, 1) = 0, I{q,YyYB) = 0, (15) 



in which y is any constant vector. 



In particular if we write q = t + p so that D becomes 



D = -,^-V (16) 



the conditions for an exact differential are 



-g7^(^.y) + SyV.i^(y,l) = 0, F{q, YyV) = 0; (17) 



and for a scalar iategral, or if F{q, dq) = ^pdq, the conditions 

 reduce at once to 



~Yp + V^p = 0, YYYp = (18) 



at 



Art. 4. — When the variable is a vector p, equations (2) and (11) 

 become 



Q = p F{p,dp); Q, = Q + JiF{p,YYYdp8p) (19) 



JPo 

 because on replacing q ^j p and D by - V we have 



dq&SqD - Sq&dqD = - dpSSpV + 8pSdpV = VVVdpSp. 



The double integral is consequently taken over the surface generated 

 by the motion of the path of integration from the first to the second 

 mode of passage or path of integration. 

 Por a closed circuit 



Q = SF{p, dp) = - jjF{p, YYYdpSp) ; (20) 



and for a scalar integral we have Stokes's theorem 



Q = JSo-dp = -JJSo-YWdpSp = JJ SYVo-YdpSp. (21) 



Art. 5. — The results of Art, 2 may be extended to a class of 

 integrals not considered by Hamilton, 



Q = Hns,^s,^'s) (22) 



in which the limits are fixed while F{q, dq, d'q) is distributive with 



