EXAMPLES OF TANGENTS. 285 



Example (4). Suppose that the equation (j>(x,y) = Q, or 

 u = 0, can be put in the form 



where v n , v^, ...... are homogeneous functions of the degree 



n, n 1, ...... respectively; hence 



dx dx dx 



du _ dv n dv n _ l 

 dy dy dy 



and the equation to the tangent is 



But by the property of homogeneous functions (see 

 Example 3 at the end of Chapter VIII.) 



dv n , dv n 



y^ +x dx =nv > 



dv... dv... , ,. 



Hence the equation to the tangent becomes 



, / 

 \ 



^ 



'" 



dy dy '" \dx dx 



or, since v n + v^ + v._ 2 . . . + v t + v = 0, 



, fdv n dv^ , \ ,/dv n dv n _ \ 



y (-J- + 7-^ + ...... )+x -j- + j ^ + ...... 



1 \dy dy ) \dx dx J 



+ + 2y - + ... + n - 1 + nv = 0. 



