Mr. J. J. Sylvester on some new Theorems in Arithmetic, 18d 



will be equal to ( — 1)* multiplied by the complete sum of the 

 (2i + l)th products of the series n, — (?i— 2), (w— 4), ... ±1. 

 Thus if w=9, the two series of elements are respectively 



9,16,21; 9,-7,5,-3,1; 

 and we find 



5x1=9-7+5-3+1 



5x(9 + 16 + 21) = 230=9.7.5-9.7.3 + 9.7.1 + 9.5.3 

 -9.5.1 + 9.3.1-7.5.3 + 7.5.1-7.3.1 + 5.3.1 

 5 x(9x21)=9x 7x5x3x1. 



I now pass on to the cases where n is an even number. \ 



Third Theorem. Let n be of the form 4m + ^, where k is zero 

 or 2 ; construct the sequence 



l.», S(»-l), 3(»-2)...Q-l)(|+8); 



the fth anakolouthic series of products formed out of these ele- 

 ments is equal to the ith complete series of products formed out 

 of the elements (/i— 2)^, {n-Qf, . . . F. 



Ex, Let w=10, the two sequences will be ' '' ^ 



10,18,24,28, ,|,afe 



64, 16, 0, 

 and we have r 



10 + 18 + 24+28=80=64 + 16 



10x24 + 10x28 + 18x28=1024=16x64. 



So, if 71=12, the two sequences will be 



12, 22, 30, 36, 40 



100,36,4; 

 and we have 



12 + 22 + 30 + 36 + 40=140=100 + 36 + 4 

 12 X (30 + 36 + 40) +22 x (36 + 40) + 30 x 40 



=4144=100x36 + 100x4 + 36x4 

 12x30x40=4x36x100. 



Fourth Theorem, If n be any even number, and we form the 

 three sequences 



\.n, 2(n-l). 3(n-2)...|(|+l) 

 l.(« + 2), 2(n + l), 3(«)...^g+3) 

 \.n, 2{n-l), 3(«-2)... (^-2)(|+3), ,,^, 



