516 PROCEEDINGS OF THE AMERICAN ACADEMY. 



of currents, and expresses the laws of the phenomena perfectly with 

 no other assumptions than equations (1) —(4). Therefore, it seems 

 reasonable to suppose that this same one principle may replace equa- 

 tions (5)-(10), and reduce the number of necessary laws from 10 to 5. 

 This fundamental principle is Hamilton's Principle, which says that 

 for any dynamical system whose kinetic and potential energies are 

 T and W respectively, 



u 



8 j {T—W)dt= 0, 



where h and ^2 are any two times, and where the variation from the 



actual motion is any variation, consistent with the constraints of the 



system, that makes the configurations of the system at the times h 



and ti the same as it is in the actual motion. In the case of the ether, 



+ - + - 



writing E for E + E and H for H + H, this principle takes the form, 



(11) 5 r r r A (H- - gw) - (e^ - ^e-^ + 2 u) ] drdt = 0, 

 ti 00 



where U is the sum of the hydrostatic tensions in the positive and 



negative electrons, if any, in which the element f/r lies, and which 



+ — 



produce the forces K and K, and where two configurations are to be 



+ - 



considered the same if, and only if, the vectors E and E are the same 



in one as in the other.* 



To prove that equations (5)-(6) result from equations (l)-(4) and 



equation (11) we may write (11) in the form, 



ti 



(12) / / / r{(H-5H— 6'H.5H)— (E.5E— GE.5E+5f/)}(/7(/i=0, 



+ 



and then suppose that 5H, 5E, 5E, and bU are all zero throughout the 



+ 



interval. Now whatever vector H may be we may split it into a sum 



+ + 



of two parts, H^ and H^,, such that 



V-H5 = = VxHl, 



4 For another form of Hamilton's Principle, involving different assumptions 

 see Larmor, "Aether and Matter," Chapter I. 



