326 Observations on the Solution 

 yi^M-]-Ll+-^),OTy= -~. By adding this to /' 



M + L.h+ 



bLb 



Mb+j^ + V'a-V'b 



we shall have x = l -\- 7/= — . If we make use 



^ M- 



of hyperbolic logarithms, then M and M^ are each equal to 1, 

 But if we use the common tabular logarithms, M = •4342945, 

 and M^ = '1SS61I7. 



X 



Example. Let x^= 100, and b = 2-2. Taking the hyperbolic 



, .^, 2-2 + l-'2682993+l-5\27\SQ2—l-76'2'sa3^ 5-2331563 



loeanthms x= = = 



° 1 + -7884574 + -5764996 2-3649570 



2-21279 + . True to the last figure, 2-21280 being rather nearer 

 the true result, but too great. 



It is only of these two orders that I intended to speak ; but I 

 cannot refrain from pointing out the direct solution of the pro- 



&c. 



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blem, when the order is infinite. Let then x =g; or we may 



&c.) 



( * 

 write it thus, x =a. But all \vithin the parenthesis is also 



equal to a; whence x'^ = a, and x=:o^. This, though at first 

 sight the most difficult of all, becomes thus the most simple. 



We now come to the third class ; but have said so much on 

 the last, that it would be superfluous to give more than the two 

 following modes of solution ; one bv infinite series, and the other 

 by an approximating simple equation. 



Let then x = a^. Bv substituting in place of the exponent x, 



its value a", we have .r = a* , and by continued substitutions of 



&c. 



the value of x, we obtain the simple series o* =.r. This, 

 however, is a very useless series : we shall therefore give the other 

 mode, which will be found much more convenient. 



Let then x^=a, and -- = La. Let x = b + y, then La = 



-J — ( L^'+Ll + -f )= ^ h T7-r-r "early, = ,, , — , and 



l + y\ ^ b/ b + y {b+y]b •' b^" + by 



y(M — bLa)=b-La — bH'; and thence ?/= —r^^—: from 



which x = 0+y=:---r———. 



This formula may be more easily deduced by the considerar 



tion, that if j:*=a, then (~^ = — : if then — = z, then z^ = 



I 



