Make the icing: In a saucepan combine the sugar, corn syrup, 1/3 cup water, bring the mixture to a boil, covered, over moderate heat, stirring occasionally to dissolve the sugar, and boil the syrup, uncovered until it registers 240F on a candy thermometer. While the syrup is boiling, in a heatproof bowl with an electric mixer beat the whites with a pinch of salt and the cream of tartar until they are frothy. As soon as the syrup reaches 240F. add it to the whites in a thin stream, continuing to beat the whites while the syrup is being added. Beat in the vanilla and beat the icing until the bowl is not longer hot. (If the icing is too stiff, beat in 1 to 2 tablespoons hot water, or enough to form a fluffy, spreadable icing.) Spread the icing over the side and top of the cake and cover the cake with the coconut. Garnish the cake with the lime slices.