1. Make the cake: Line the bottoms of 2 greased 8 by 2 inch round cake pans with rounds of wax paper and grease and flour the pans, shaking out the excess flour.

2. Spread the coconut in a jelly-roll pan, toast it in a preheated 350F oven, stirring once or twice, for 7 minutes, or until it is golden and let it cool.

3. In a bowl whisk together the flour, the baking powder, the salt.

4. In a large bowl with an electric mixer cream the shortening, beat in 3/4 cup of the sugar, and beat the mixture until it is light and fluffy.

5. Beat in the yolks, one at a time, and the vanilla and add the flour mixture alternately with the milk, beginning and ending with the flour and beating until the batter is smooth.

6. In another large bowl with the electric mixer beat the whites with the salt until they are foamy, add the cream of tartar, and beat the whites until they hold soft peaks.

7. Beat in the remaining 1/2 cup sugar, a little at a time, and beat the meringue until it holds stiff glossy peaks.

8. Stir! the toasted coconut and one fourth of the meringue into the batter to lighten it and fold in the remaining meringue gently but thoroughly.

9. Divide the batter between the prepared pans and bake the layers in the middle of the preheated 350F oven for 30 to 35 minutes, or until tester comes out clean.

10. Let the layers cool in the pans on racks for 10 minutes, invert them onto the racks, and let them cool completely.

11. Make the filling: In a small heavy saucepan whisk together the sugar, cornstarch, salt, lime juice, lemon juice, and the egg until the mixture is combined and bring the mixture to a rolling boil over moderately high heat, whisking constantly.

12. (The mixture will thicken as it cooks.

13. ) Whisk in the butter and let the filling cool.

14. Split each cake layer in half horizontally with a long serrated knife and brush off any loose crumbs.

15. On a cake plate arrange one of the cake layers, cut side up, and spread it with one third of the lime filling, leaving a 1/2 inch border.

16. Sprinkle the filling with 1/4 cup of the coconut and top the coconut with another cake layer, cut side down.

17. Continue to layer and fill the cake in the same manner with the remaining filling and coconut.

18. 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.

19. 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.

20. As soon as the syrup reaches 240F.

21. add it to the whites in a thin stream, continuing to beat the whites while the syrup is being added.

22. 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.

23. ) Spread the icing over the side and top of the cake and cover the cake with the coconut.

24. Garnish the cake with the lime slices.