190 Mr. J, J. Sylvester on some new Theorems m Arithmetic. 



the tth anakolouthic sum in respect to the second sequence less 

 the ith anakolouthic sum in respect to the first sequence is equal to 



9(9 "^ ^ ) ^^*^ *^^ («— l)th anakolouthic sum in respect to the 



third sequence. ^ 



Ex. Take the three sequences 



1.10 2.9 3.8 4.7 5.6 



1 .12 2.11 3.10 4.9 5.8 



1.10 2.9 3.8. 



These, written out with simple elements, are as follows : — 



10 18 24 28 30 



12 22 30 36 40 



10 18 24; 

 and we have 



(12 + 22 + 30 + 36 + 40)-(10+18-f24 + 28 + 30)=30xl 

 {12 X (30 + 364-40) +22 X (36 + 40)4-30x40} 

 -{10(24 + 28 + 30) + 18(28 + 30) +24x30} 

 =4144-2584=1560=30 X (10 + 18 + 24) 

 12 . 30 .40-10 . 24 . 30=14400-7200 = 7200 



=30 X (10x24). 

 These four theorems are only particular cases of one much 

 more general relating to a determinant, to which I was led by my 

 method of integrating the system of two partial differential equa- 

 tions to the general invariant of a function or system of functions 

 of two variables. In like manner the integration of the system of 

 t partial differential equations to the general invariant of a func- 

 tion or system of functions of t variables conducts to a determi- 

 nant*, of which a priori we know the constitution, and which 

 will (save as to the periodic occurrence of a single factor X) 

 resolve itself into factors of the form X' + m*, m being an integer, 

 and thus promises to lay open a road to the discovery of a new 

 genus of theorems relating to the powers of the natural pro- 

 gression of integer numbers, destined apparently to occupy a 

 sort of neutral ground between the formal and quantitative 

 arithmetics. 



25 Lincoln's Inn Fields, 

 July 11, 1864. 



* The integration of this system of equations always depends essentially 

 upon the integration of one homogeneous equation which is doubly linear, 

 t. e. of the first degree in the variables, and also of the first degree in 

 respect to the order of the differentiations ; such an equation can always be 

 integrated, and the integral will depend upon the solution of an algebraical 

 equation expreised by equating a certain determinant to zero. 



