254 Compounds Isomeric with the Sulphocyanic Ethers. [Feb. IB, 



between that and the next term are found by the processes mentioned in 

 the former part of this paper. The roots of the even numbers are then 

 obtained by an analogous process ; and these are used as bases or roots of 

 the polygonal numbers, which are placed in columns, with their sums, as 

 appears in the Table (see Diagram No. 4 for the mode in which the poly- 

 gonal numbers are formed). 



It will be observed that the sum of the roots or bases is 1 7 ; but if they 

 be used to form trigonal numbers, the increment of the sum of the re- 

 sulting trigonal numbers above the sum of the roots or bases is 28, and so 

 on of the rest, each successive column increasing by the same number, viz. 

 28. If the roots or bases be n, n,n+\ (that is, a term in the gradation 

 series), the increment of the sums of the successive columns will be 

 2/i^ip?2, a trigonal number. 



Again, in the trigonal numbers the difference between the sums of the 

 first and second term is ; in the square numbers it is 1 ; in the pentagonal 

 numbers 2 ; in the hexagonal numbers 3 ; in the heptagonal numbers 4 ; 

 but in all of them the difference between the second and third terms is 1, 

 and this continues throughout. The difference between the third and 

 fourth, the fifth and sixth, the seventh and eighth, &c., increases by 1 in 

 each column ; but the difference between the second and third, the fourth 

 and fifth, the sixth and seventh, &c., is always 1 in each column ; and the 

 result is that, by adding 1 in the pentagonal column, by adding 1, or 1 . 1 

 in the hexagonal, by adding 1, or 1 . 1, or 1 . 1 . 1 in the heptagonal, every 

 number, odd or even, can be made by not exceeding four square numbers, 

 or five pentagonal numbers, or, &c., as clearly appears by the Table. 



This corresponds with what was discovered by Cauchy, published at the 

 end of Legendre's ^Theorie des Nombres,' viz. that four only of each class 

 of numbers is necessary ; the rest may be supplied by 1 repeated as often as 

 necessary. But I must not omit to say that, although all the odd numbers 

 are sufiiciently obedient, there is one class of even numbers quite refractory, 

 viz. the powers of 2. They maybe easily expressed in squares, pentagonal 

 numbers, &c., but they cannot be brought within the rule that otherwise 

 prevails. 



II. " Compounds Isomeric with the Sulphocyanic Ethers. — I. On the 

 Mustard Oil of the Ethyl Series.^' By A. W. Hofmann, LL.D., 

 F.R.S. 



The results of my researches on the chloroform-derivatives of the 

 primary monamines, which, as I have shown, are isomeric with the nitriles, 

 could not fail to direct my attention to allied groups of bodies, with the 

 view of discovering similar isomerisms. 



In a note communicated to the Royal Society some months ago, I 

 expressed the expectations which even then appeared to be justified in the 

 following manner : — " In conclusion, I may be permitted to announce as 



