lOi On Functional Equations. 



the two cases. We are not prepared, however, to follow 

 Mr. Leslie in his entire rejection of this application of analysis. 

 It seems to us, on the other hand, that the equation furnished 

 by superposition or expei'iment ought to contain all the truths 

 which are deduced from the same experiment by geometrical 

 reasoning. We are accordingl}' rather inclined to search for 

 the cause of the difficulties that have startled us, in some im- 

 perfection in the mode of expressing or of treating the equa- 

 tions. That imperfection seems to me to be a deficiency in 

 the original equation as given by Legendre. Superposition 

 does not inform him that the vertical angle is determined by 

 the base and its adjacent angles alone, — but merely that it will 

 be constant if they are constant ; and hence that it must be 

 determined by these variables and coNSTAN'rs alone. Noting 

 these constants by y, the original equation becomes 

 C=^(c,A,B,y) 

 The distinction drawn by Legendre, &c. betwixt lines and 

 anoles, will furnish us with the analysis ofy. It must, if any 

 quantity at all, be composed either of constant lines or con- 

 stant angles. It cannot be composed of constant lines, for 

 there are no such quantities in existence. But there is a con- 

 stant ano-le ; y may therefore be a function of the right angle. 

 It is not therefore a linear, but an angular magnitude. This 

 is enough to entitle us to reject the foregoing equation as im- 

 possible, and to adojit as the true one 

 C = ?s(A,B,y) 

 It is likewise enough to show that C cannot be rejected from 

 Mr. Leslie's equation, which then becomes 



as it ought to be. 



If Baron Maurice will return to the subject and contem- 

 plate it under this aspect, he will find, I trust, all difficulties 

 dismissed, and instead of the puzzling equations 



cos C = (f. C 



arc 



he will have 



and anMe = ,. 



"^ radius 



COS C = <r -1-7- r 



^ right angle 



and 



angle nrc 



right angle radius 



expressions in as complete accordance as he could wish with 

 the universal law of homogeneity. 



I meaiit to have taken this opportunity to make some ob- 



scrvations 



