in the Calculus ofFunctions, noticed hy Mr. Babbage. 335 



and this solution satisfies the equation -i^; x ■=■ c^ x"^ independ- 

 ently of any particular value of w, and if we suppose n = —l, 



— log-2 X 



we have ^x = c '°s^^ (4.) 



for the solution of the equation -^x = c^ — , whatever may 



be the value of c, and we have before shown that it cannot 

 have a solution unless c = + 1. The only explanation I am 

 at present able to offer concerning this contradiction is one 

 which 1 hinted at on a former occasion, viz. that if we sup- 

 pose ^ to represent any invei'se operation which admits of 

 several values, then if throughout the whole equation we al- 

 ways take the same root or the same individual value of •>!/, it 

 is impossible to satisfy the equation, but if we take one value 

 of \J/ in one part, and another of the values of -i^ in other 

 parts of the equation, it is possible to fulfill it by such means. 

 This solution may, perhaps, appear unsatisfactory ; it is how- 

 ever only proposed as one that deserves examination, and I 

 shall be happy if its insufficiency shall induce any other per- 

 son to explain more clearly a very difficult subject." 



This passage is referred to by Professor De Morgan in his 

 article on the Calculus of Functions, §. 72, in the Encyclo- 

 jpcedia Metropolitana, and I generally agree with him in the 

 cause which he suggests for the explanation of the difficulty, 

 viz. a dhcontinidty in the form of vf/, by which I mean that for 

 different sets of continuous values ofa;, \I/has partial non-con- 

 current solutions of different forms. What I now propose is to 

 point out specifically the existence of that cause in the in- 

 stance put forward by Mr. Babbage, and the precise rationale 

 of its application in explaining the difficulty in question. The 

 accomplishment of these objects, which do not seem to have 

 been hitherto satisfactorily effected, would not only be inter- 

 esting for its own sake, but would, in many analogous cases 

 be conducive to clearness of reasoning, and tend to restore 

 that confidence in the result of mathematical principles which 

 is impaired when processes apparently legitimate lead to con- 

 tradictory conclusions. If, according to the explanation 

 thrown out by Mr. Babbage, (as I understand it) ■^ x had 

 many values for a given x, and we were allowed to shift its 

 different values among the equations (1.) and (2.), substitut- 

 ing one of those values for the symbol v(/ x in one part of 

 either of those equations, and another in another, it is easy 

 to see that we might get rid of the difficulty of supposing 

 c* = 1, and might even, if we had a sufficiently indeterminate 

 4/x, allow toe an infinite number of values. By this plan, 



for instance, if ^x had two values, v and t/, -p- might 



