167 



the direction of x" coinciding with that o{ p, but the direc- 

 tion of y" being perpendicular thereto. Hence, 



y" _ xy' — yx 



x" xx + yy' ' 

 that is, 



tan-' ^„ =: tan"' ^ — tan-' ^ ; 



XXX 



or, finally, 



f{v'-v)=f{v')-f[v), (A) 



at least for values of z', v' , and v' — «, which are each greater 



77" 



than 0, and less than — ; ify be a function so chosen that the 

 equation 



\ = tan/(^) 



expresses the sought law of connexion between the ratio 

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



f{mv) = mf{v) - -f[nv), 



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

 components gives evidently 



hence 



/U 



/ir^ 



TT 





4' 





m 



TT 



n 



4' 



and ultimately. 



f{v) = V, (b) 



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



while V increases from to -, the function f(v) increases 



therewith, and therefore could not be equal thereto for all 



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



4 



property also for all intermediate incommensurable values. 

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

 forces X and y, the equation 



