OF ALGEBRAIC QUANTITIES. 353 



then /x\*" becomes /'l^"' 



(.)-■ becomes (1)-' = (1)" 



&c. becomes &c. 



•. A = /r ~ 



(J) 



B = „ . (■)■ 



^ — 1.2 

 &C. = &C. 



cy 



m , ™ r, I I OT.OT — 1 2,0 ") 



/2\ = (1) . I 1 + Wi -^ H f- 2— ■» + &C.| ; 



3 c 



Next, let d be greater than -t" 

 In this case, when x = 0, 



fz\"^ becomes f 1 ^ ; 





42. Let /E\ (, ) = f , where x is a positive quantity, and m a pos- 



sible quantity, and let x and g vary whilst m remains constant ; then § will 

 trace out a logarithmic spiral, which cuts its radii vectors at an angle = m. 



For let r be a positive quantity in length = g, 



then, since /E\'"'»' = /1\ c = cos m + sin m . ^ — \, 



(■r^\X COS m + X sin m . »/ — 1 

 ,.„_, isinm 



= (?) • (!)- 



f costn 

 r =: 



/EN , and gi is inclined to unity at an angle = x . sin m, 



.'. r' = a; . cos m, 







MDCCCXXIX. 2 Z 



