Mr. Graves on the Porismalic Arra?igement of Points. 129 



means, may in his present state of being always remain be- 

 yond the compass of his knowledge and understanding; 

 aIthou<;h, seeinj; through a glass darkly, he be allowed to ob- 

 tain partial glimpses ol the truth, while engaged in the de- 

 lightful task of endeavouring to trace the footsteps, and pene- 

 trate the benevolent designs, of the Creator, as manifested in 

 His wonderful and infinite works. 



XXIV. On the Functional Symmetry exhibited in the Nota- 

 tion of certain Geometrical Porisms, when they are stated 

 merely with reference to the Arrangemeyit of Points. By 

 John T. Graves, Esq., M.A., F.R.S., of the hmer Temple.'^" 



Theorem I. T ET 1, 2, 3, T, 2\ S\ V\ 2", 3'' denote 9 points. 

 ^-^ In the following scheme, let a set of 3 points 

 1, 2, r lying i7i directo be denoted thus, 1 2 T. Then, if any 

 8 of the following sets be given, the remaining set will poris- 

 niatically follow. (See fig. I.) 



\2V 1'2M" 1"2"1 



2 3 2" 2' 3' 2'^ 2" 3^' 2 



3 1 3' 3' r 3" 8" 1" 3 



The preceding theorem in all its generality may be derived 

 as a functionally formal consequence, if we assume the two 

 following propositions: 1. The order of succession of the 

 given sets is immaterial. 2. There is a certain system of 8 

 sets out of the preceding 9, from which the remaining set will 

 porismatically follow. The former proposition may be taken 

 as a postulate from the nature of the subject which the sym- 

 bols represent. In fact, the order of succession of the sym- 

 bols in any single set is also immaterial. The second proposi- 

 tion requires geometrical proof. Assuming the former, it is 

 easily seen that any one system of 8 sets may be put into the 

 same form, with respect to its constituent symbols, as any other 

 system of 8 sets, and any remaining set is formally derived 

 from any one such system in the same manner as any other 



* Communicated by the Author. 

 Phil. Macr. 6. 3. Vol. 15. No. 9i. Aug. 1839. K 



