32S Observations on the Solution of Exponential Equations. 



a is therefore equal to "6922. If a be less than this, no solution 

 can take place. This result might be got easier by considering 

 that if a be a minimum, then La is also a minimum ; whence 

 xhx is a minimum, and the ratio of the differentials or 



■ j^ =u, and riLx= — 1, and so on. 



We may take one more example, which exhibits the limita- 

 tion in a very striking manner. It is when x^ = w. for then 

 "■^—^ = 0, and HLa:=l. Whence a- = 2-71S28, and (2 = 



1'4447, when it is a maximum: that is, if a exceeds this num- 

 ber the question is impossible. 



I have now closed mv remarks on these equations, still, I am 

 convinced, but very imperfectly handled. Many more observa- 

 tions might be added, and indeed the higlier orders especially 

 require a far more ample investigation, and at some future pe- 

 riod I may perhaps offer a few more remarks upon them. In the 

 mean time I may only observe, that from the short glances I 

 have cast over them, it is my opinion that to obtain these solu- 

 tions, formulae of a different kind from those I have as yet em^ 

 ployed must be made use of, and which, I have no doubt, may 

 lead the inquirer to some interesting speculations. 

 April 3, 1817. 



To Mr. Tilloch. 

 Sir, — Since sending you a few days ago some modes of solu- 

 tion regarding exponential equations, another mode (and an ex- 

 tremely simple one) has occurred to me, the insertion of which 

 along with the other part would much oblige, sir. 



Yours &c. 

 Edinburgh, April 20, 181?. G. A. WaLKER ArNOTT. 



The mode which I now offer is founded on the principle of 

 Taylor's theorem ; it is capable of greater extension than the 

 former modes, as we can apply it, with some address, to any 

 order. 



Taylor's theorem is this : that if ?/ be a function of z, and z 

 receive an addition c, y then becomes equal to 



3^+^ i + k: • .r^ + w-s • d + ^''^- T« ^i^p^y ^'"' 



theorem to the subject in question, the two first terms of the 

 series are quite sufficient, and we may take an example of the 

 best mode to use it. 



1st. Let ^^" = 0, and let I he an approximate value oi x, and 

 correspond to y in Taylor's theorem. Let also bLb, which is 

 evidently a Junction of b, correspond to z : then if bLb receive 

 an addition Cj so that bLb + c=Laf ov c=La—bLb, the two 



first 



