\(u 



the direction of a" coinciding with that of/;', but the direc- 

 tion of y" being perpendicular thereto. Hence, 



that is, 



x" xx + yy' ' 



tan-y„=tan-^-tan-'^; 



X A -t 



or, finally, 



/(,/_,;) =/(^-')-/(^0, C'^) 



at least for values of r, v' , and »'- », which are each greater 

 than 0, and less than ^ ; if/ be a function so chosen that the 



equation 



■L - tan/(i;) 



expresses the sought law of connexion between the ratio 

 ^J- and the angle v. The functional equation (a) gives 



V 



f{mv) = mfiy) - -f{nv), 

 m and n being any whole numbers ; and the case of equal 

 components gives evidently 



hence 



and ultimately. 



Jin 7r\ _ m TT 



f{v) = V, («) 



because it is evident, by the nature of the question, that 

 while V increases from to ^, the function y(r) increases 

 therewith, and therefore could not be equal thereto for all 

 values of v commensurable with ^, unless it had the same 

 property also for all intermediate incommensurable values. 

 We find, therefore, that for all values of the component 

 lorces X and y, the equation 



