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XXX. 



SOME APPLICATIONS OP HAMILTOIs^'S OPEEATOR V IN 

 THE CALCULUS OF YARIATIONS. By CHARLES J. 

 JOLT, M.A., F.T.C.D., Andrews' Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 



[Read December 11, 1899.] 



In cases similar to those treated by Tait (Quaternions, 3rd ed., p. 403), 

 we may throw the integral into the form ^fdp, /( ) being a linear and 

 distributive function of a vector. The conditions for a stationary 

 value of the integral become /Fv VdpSp = over the curve, and 

 fSp = at the limits. In the first of these equations v operates on/ 

 alone and in situ. 



Again for surface integrals of a somewhat similar type, the condi- 

 tions for a stationary value of the integral JT FVdpcl'p are i^v = over 

 the surface, and F\ = over the bounding curve. Here, as before, 

 V operates on the linear and distributive function F alone and in situ, 

 and X is the normal to a given surface upon which the boundary is 

 constrained to lie. 



Surface integrals of the type ^juTVdpd'p are reduced to this form 

 by writing Uv = UVdpd'p and TVdpd'p= - SUvVdpd'p. Observing 

 that ^v Jjv = - {Ki + Ko) where Ki and K^ are the principal curvatiires 

 at the point, i^v = becomes in this case 



du , -^ .^^ 



J + (Xi + K.)u = 0. 

 dn 



