ential ▁systems ▁Cart an ' s ▁methods ▁in ▁the ▁theory ▁of ▁differential ▁systems ▁are ▁perhaps ▁his ▁most ▁prof ound ▁achiev ement . ▁Bre aking ▁with ▁tradition , ▁he ▁sought ▁from ▁the ▁start ▁to ▁form ulate ▁and ▁solve ▁the ▁problems ▁in ▁a ▁completely ▁invariant ▁fashion , ▁independent ▁of ▁any ▁particular ▁choice ▁of ▁variables ▁and ▁unknown ▁functions . ▁He ▁thus ▁was ▁able ▁for ▁the ▁first ▁time ▁to ▁give ▁a ▁precise ▁definition ▁of ▁what ▁is ▁a ▁" general " ▁solution ▁of ▁an ▁arbitrary ▁differential ▁system . ▁His ▁next ▁step ▁was ▁to ▁try ▁to ▁determine ▁all ▁" sing ular " ▁solutions ▁as ▁well , ▁by ▁a ▁method ▁of ▁" pro long ation " ▁that ▁consists ▁in ▁ad jo ining ▁new ▁unknown s ▁and ▁new ▁equations ▁to ▁the ▁given ▁system ▁in ▁such ▁a ▁way ▁that ▁any ▁singular ▁solution ▁of ▁the ▁original ▁system ▁becomes ▁a ▁general ▁solution ▁of ▁the ▁new ▁system . ▁Although ▁Cart an ▁showed ▁that ▁in ▁every ▁example ▁which ▁he ▁treated ▁his ▁method ▁led ▁to ▁the ▁complete ▁determ ination ▁of ▁all ▁singular ▁solutions , ▁he ▁did ▁not ▁succeed ▁in ▁prov ing ▁in ▁general ▁that ▁this ▁would ▁always ▁be ▁the ▁case ▁for ▁an ▁arbitrary ▁system ; ▁such ▁a ▁proof ▁was ▁obtained ▁in ▁ 1 9 5 5 ▁by ▁Mas ata ke ▁Kur an ishi . ▁ ▁Cart an ' s ▁chief ▁tool ▁was ▁the ▁calculus ▁of ▁exterior ▁differential ▁forms , ▁which ▁he ▁helped ▁to ▁create ▁and ▁develop ▁in ▁the ▁ten ▁years ▁following ▁his ▁th esis , ▁and ▁then ▁proceeded ▁to ▁apply ▁with ▁extraordinary ▁virt u os ity ▁to