100 Mb murphy, ON THE RESOLUTION OF 



(Q-ef i^-ey (rey {2-'6y _ ^^ 



"'"1.2 ''"1.2.3.4 1.2.3.4.5.6 1 . 2...8 



= COS. (2'0), 



and the continued surd which is the inverse function may be similarly 

 expressed. 



a . a 



a .a 



III. To expand the function a. a, the indices being continued 

 X times. 



Denoting this quantity by «^, we have 

 !<.,+ , = a ■ a"' = ^(«x), 

 and adapting the general formula; to this case, we have 

 A = a. a'\ whence A is known, 

 7 = ^ . h.l. (a), 



A(h.l.ay 

 1 .2 



&c. = &c. ; 

 and then u, = A + By' + c, {By')"' + c, {Byy + &c., 

 % = A + B + c,B' + c,B' + &c. 



For the inverse function if we put « = ^ . and take logarithms in 

 the system of which the base is a, we have 

 u, = log. fi + log. ?<,+ , = log. (/3?<.,+ ,). 



M_, = log. {/??<_ (,_,)}, 



«_, = log. \ft . log. {iSlog. /3 log./3«J, d- times ; 



Ci7' = 12 + '^iT^ 



