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XXXI. — Solution of a Functional Equation, with its Application to the Parallelo- 

 gram of Forces, and to Curves of Equilih^ation. By William Wallace, 

 LL.D., F.R.S.E., F.R.A.S., M. Camh. Phil S., Hon. M. Inst. Civ. Engin., 

 Emeritus Professor of Mathematics in the University of Edinburgh. 



(Read 2d December 1839.) 



Article 1. The introduction of the notion of 2, function of a variable quantity 

 into the mathematics, without any regard to its particular form, has given vast 

 extension to the science, and been the germ of some of its most important theo- 

 ries. The doctrine of curve lines, no doubt, produced that of functions, for the 

 former may be made the visible expression of the latter : thus, either of the co- 

 ordinates of a curve being taken as the representation of the variable, the other 

 co-ordinate is 2, function of the variable ; so also are the arc of the curve, and its 

 area. Indeed, in contemplating functions, and discussing their properties, it is 

 convenient to substitute in our reasonings the geometrical representation for the 

 abstract notion of the function. 



2. In addition to the aid which geometry gives us in forming distinct notions 

 of the relations of functions, the notation of modern analysis affords farther as- 

 sistance in discussing their properties. In our Trigonometrical Tables, the co- 

 sine, sine, tangent, &c., are all regarded as functions of the angle ; and the calcu- 

 lus of sines is, in fact a creation of the mind, called into existence by the power 

 of a few abbreviations of the words sine, cosine, ^c, which, as symbols, serve in 

 our processes of reasoning to represent the things they signify. 



3. From the notion of a function, which we acquire from geometrical exten- 

 sion, combined with the use of the arbitrary symbols of analysis, we learn that it 

 presents to the mind two distinct objects ; namely, its form, and its properties : 

 for example, the function log x, that is the logarithm of a number x, may be re- 

 presented geometrically by the ordinates of a curve ; also, by spaces between a 

 hyperbola and its asymptote ; these ordinates and spaces have certain relations 

 to each other, which are the properties of the function. Among its analjrtical pro- 

 perties there is this one, 



log a; + log ^= log (a;^) , 



which is deducible from this definition of the function, that it is the exponent of 

 the power of some given number ^ which number being raised to that power., pro- 

 duces X. 



VOL. XIV. PART II. 5 Z 



