Rev. T. P. Kirkmau's New Theorems in Combinations. 209 



improper to suggest that the remaining manusex'ipts ought to 

 be deposited in some public library, where they would at once 

 be safe and accessible, and like Dr. Simson's Adversaria at Glas- 

 gow, ever remain an enduring monument of the genius and in- 

 dustry of so devoted a geometer. 



Burnley, Lancashire, 

 June 17, 1852. 



XXX, Theorems in the Doctrine of Combinations. By the Kev. 

 Thomas P. Kirkman, A.M., Rector of Croft with Southworth. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



I WOULD beg your permission to enunciate the following 

 theorems in your Joiu'nal : — 



A. With 7 symbols can be formed 21 triads, so that every 

 duad shall be thrice employed. 



B. Two distinct systems of 7 quadruplets each can be made 

 with 7 symbols, both exhibiting twice all the 21 duads. 



C. A system of 21 quadruplets can be made with 7 symbols, 

 so that every possible duad shall be six times employed, 



D. With 13 symbols can be made three different groups of 

 triads, each group once containing all the duads. 



E. With 15 symbols different triads can be made, so as to 

 exhaust the possible duads, once, twice, 3, 4, 5, 6, 7, 8, 9, 10, 

 11, 12, or 13 times. 



F. With (12/1 + 3) symbols can be formed tiiads so as to ex- 

 haust the duads 6n — 1, or 6« + 2 times; and with 12n + 7 sym- 

 bols, so as to exhaust the duads 6n+ 1, or 6^ + 3 times. 



G. With 27 symbols triads can be made, till the duads have 

 been all twice employed, or all thrice employed. 



H. With 4(3n + 1) symbols quadruplets can be made, till every 

 duad has been (2w + 1) times employed, and this without repeat- 

 ing any triplet. 



I. With 4x2" symbols, quadruplets can be made till every 

 triad has been once employed, 



J. Sixteen young ladies can all walk out four abreast, till every 

 three have once walked abreast ; so can thirty-two, and so can 

 sixty-four young ladies ; so can 4" young ladies. 



Croft Rectory, near Warrington, 

 August 6, 1852. 



Phil. Mag. S. 4. Vol. 4. No. 24. Sept. 1852. 



