Based　on　the　solid　mechanics　of　the　discrete　form　and　its　variational　principles　proposed　by　Niu[1-2],　this　paper　puts　forward　four　kinds　of　boundary　integral-variational　theorems　of　an　arbitrary　element.　In　the　course　of　the　fracture　analysis,　they　can　be　used　to　compute　the　energy　release　rate　along　the　normal　direction　of　the　crack　boundary.　When　there　is　a　hole　in　the　solid,　and　whether　there　are　given　surface　forces　on　the　hole　boundary　or　not　they　can　be　used　to　compute　the　variation　of　the　energy　along　the　normal　direction　of　the　hole　boundary.　In　the　course　of　the　discrete　analysis,　they　can　be　used　to　establish　the　discrete　equations,　so　that　the　values　of　the　unknown　functions　are　solved.　At　the　same　time,　from　this　paper　we　know　that　the　J-integral　proposed　by　Rice[3]　represents　an　integral　to　be　independent　of　a　path　imperfectly.　©　1984　Hust　Press.