201 Mr. T. Graham on Liquid Diffusion applied to Analysis. 



above written. {Vide a paper of mine "On the perfect ^-parti- 

 tions of N = r 2 — r + 1/' Transactions of the Historic Society of 

 Lancashire and Cheshire, vol. ix. 1857.) 



For example, the perfect sets of triplets for N = 13 arc ob- 

 tained by solution of the six congruences, made with six values 

 of m, 



a L m + a™ + a 3 m + b{" + bj» + b 3 m + («, + aj>» + [a % + a 8 ) m + {a 3 + a k ) m 

 + (b l + bo) m + (b Q + b 3 ) m +(b 3 + b l ) m = l m + 2 m + .. + 12m, (mod. 13,) 

 of which one solution is 



a 1 ff 2 a 8 +& 1 ft 2 & 8 =139.+265. 

 Perfect partitions are the following : — 

 N = 7, 1.2.4: N = 13, 1.3.9 + 2.6.5: N = 19,1.7.11 



+ 2.14.3 + 4.9.6: 

 N = 13, 1.2.6.4; 1.7.2.3: N =21, 1.3. 10. 2.5 : 

 N,=31, 1.5.12.4.7.2: 



N = 73, 1.2.4.8.16.5.18.9.10; 1.4.7.6.3.28.2.8.14; 

 1.16.22.23.4.6.8.11; 1.8.12.2.3.13.24.4.6. 



Using 139 + 265, or 1264 as difference circles, we form the 

 triplets 1, 2, 5 +4, 6, 12, or the quadruplet 1, 2, 4, 10. If we 

 now complete under every element the circle 12345 ... 13, we 

 shall have two columns of triplets, or a column of quadruplets, 

 in which the duads are once, and once only exhausted. In the 

 same way all the above perfect partitions may be used. 



Six solutions of the triplet problem for 12w + 3 = 27 are given 

 by Mr. Anstice. They are all derangements of the same group 

 of the thirteenth order. The solution of the same problem by 

 Mr. Mease (Camb. and Dub. Journ. vol. v. 1850, p. 262) is the 

 sum of certain derangements of two groups of the third and 

 ninth order added to a derangement of unity. 



The method above given, depending on the theory of differ- 

 ence circles, will exhaust the solutions of Mr. Anstice's form, 

 and can easily be modified so as to exhaust those of Mr. Mease's. 



Croft Rectory, near Warrington, 

 January 14, 1862. 



XXXI. Liquid Diffusion applied to Analysis. 

 By Thomas Graham, F.R.S., Master of the Mint*. 



THE property of volatility, possessed in various degrees by so 

 many substances, affords invaluable means of separation, as 

 is seen in the ever-recurring processes of evaporation and distilla- 

 * From the Philosophical Transactions for 1861, Part I. p. 183. 



