176 THE LAWS OP 



hibited, and comparing like powers of the indeterminate quan- 

 tity r gives 



(i) t (t) 3 n (i) 3 n.5 n (i) 3n.5n.7n (i 



C l= " 2 ^ " 2 . 4 a * ' 2 . 4 . 6 a * 



(i) = i (i), 3 ~^ fl . 3-n.5-n 



c i 1>a i 2 a 2.4 a 



&c.= ......... &c .......... &c. 



of which system the general type is 



the symbols of operation being here separated from those of 

 quantity, and e being used in its ordinary acceptation with re- 

 ference to the lower index w, so that we shall have generally 



'". <*!? = <. 



The general equation between ajf and c w being resolved, evi- 

 dently gives by expanding the binomial and writing in the 

 place of ec?, e ! c', e'c" 1 , &c. their values c&,, c. s , <$,, &c. 



Having thus the value of oj, 1 ' we thence immediately deduce the 

 value of A (i} and this quantity being known, the first line of the 

 expansion (25) evidently becomes known. 



In like manner when we suppose that the quantity EP 9 is 

 expanded in a series of the form 



; 2).r'+&c.} 



