It is well known that although the overwhelming majority of specimens of fircones exhibit one or other of the simple spiral arrangements represented by the terms of the ordinary series ½, ⅓, ⅖, ⅜, , &c., whose generating and successive secondary spirals are indicated by the numbers 1, 2, 3, 5, 8, 13, &c., yet exceptional cases occur now and again, where we find either conjugate spirals of the ordinary system, or arrangements (usually simple, but sometimes conjugate) belonging to other systems of spirals. Of these exceptional arrangements, perhaps the most common are bijugates of the ordinary system, giving the numbers 2, 4, 6, 10, 16, 26, &c., and simple spirals belonging to the system ⅓, , &c., giving the numbers 1, 3, 4,7,11, 18, &c. More rarely, trijugates of the ordinary system occur, giving the numbers 3, 6, 9, 15, 24, 39, &c.; or spirals of the system , ⅕, , &c., giving the numbers 1, 4, 5, 9, 14, 23, 37, &c.; not to speak of various other arrangements, some of which will fall to be considered in the special cases which form the subject of the present communication.