MAJOR P. A. M \.M.\HON ON THE COMPOSITIONS OF NUMBERS. 103 



as denoting 



i 



,...Mi- 1* t ...r. + "*...-,, P.+1,, f,...fir,...r. 



and 



as denoting 



according to the law usual in this subject ; we find 



"ft ...p,hq, ...jAv- r. "">,... p.. ,,*+7i. tt-lfl'i ' 

 ''-- "' - 



< 

 = fa { (Pi 



= <M (pi. 



= hMk 



The reader may verify that 



*{(P) (9) W}. 

 have each the same value W ,_ 



which we have denoted by , 



"rv 



Art. 60. I pass on to drag into the light some important relations connecting 



h Pl ...r, and ,,.... 

 When the relation 



/, - V/V+* 1 (^0' *. * 



ft, - i(-) ,-. -yt a " 

 AI :. ..,. 



was under ol>servation just now, it will not have escaped notice that this is precisely 

 the expansion of 



n = i a -2, 

 a m = a, s 2a,a 3 + a 3 ; &c., 



and by the law of formation we see that 



a,. = /< , 



and theuce 7 



a, = A,.. 



