266 THIRD REPORT — 1833. 



favour of the affirmative of this proposition, which were for the 

 most part founded upon the analytical interpretation of the pro- 

 perties of the hyperbola and logarithmic curve, were not en- 

 titled to much consideration, in as much as they were not drawn 

 from an analysis of the course followed in the derivation of the 

 symbolical expressions themselves and from the principles of 

 interpretation which those laws of derivation authorized. A 

 very slight examination of those principles, combined with a re- 

 ference to those upon which algebraical signs of affection are in- 

 troduced, will readily show the whole of the very limited nvmi- 

 ber of cases in which such a proposition can be considered to 

 be true *. 



* In the 15th volume of the Annales des Mathematujues of Gergonne, there 

 is an ingenious paper by M. Vincent on the construction of the logarithmic 

 and other congenerous transcendental curves. Thus, if y = e^^ there will be in 

 the plane of x y a continuous branch such as is commonly considered, and a 

 discontinuous branch corresponding to those negative values of y which arise 

 from values of x, which are expressible by rational fractions with even deno- 

 minators : thus, if we suppose the line between x z= and a- = 1 to be di- 

 vided into an even number 2 p of parts, (where p is an odd number,) the 

 values of* will form a series effractions, 



J_ ± 3 2 2p-l 



2p p Zp p 2p 



which have alternately odd and even denominators, and which correspond 

 therefore to values of y which are alternately single and double. If we may 

 suppose, therefore, a curve to be composed of the successive apposition of 

 points, the complete logarithmic curve will consist of two symmetrical 

 branches, one above and the other below the axis of x, one of which, in cor- 

 responding parts of the curve, will have double the number of points with the 

 other. The inferior curve, therefore, may in this sense be considered as dis- 

 continuous, being composed of an infinite number of conjugate points, forming, 

 in the language of M. Vincent, une h-anche pointillee. 'J he same remark ap- 

 plies to other exponential curves, such as the catenary, &c. 



It was objected to this theory of M. Vincent by M. Stein, another writer 

 in the same journal, that every fractional index in this interval might be con- 

 verted into an equivalent fraction with an even denominator, which would 

 give a double possible value of the ordinate, which would be different from 

 that given by the fractional index in its lowest terms ; and that consequently 

 there would necessarily be a double ordinate for every point of the axis, and 

 therefore also a double number, one positive and the other negative, corre- 

 sponding to every logarithm. In reply to this objection, it is merely neces- 



m mp m mp 



sary to observe that the values of a" and o"^ or of 1 " and 1 "^ are in every 

 respect identical with each other, the n p values in the second case consisting 

 merely of p periodical repetitions of those in the first. 



In a paper in the Philosophical Transactions for 1829, Mr. Graves has given 

 a very elaborate analysis of logarithmic formulee, and has arrived at some 

 conclusions of great generality which it is difficult to reconcile with those 

 which have been commonly received. Amongst some others may be men- 

 tioned the formula which he has given for the Napierian logarithms of 1, 



