350 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



an 



d we have also the relation, 



o^£3u'+f^^,S.Sy^^Sf (15). 



dx' dxdy dy^ 



Now we have P = (cos y — j / («) ; 



dP f d\^w s '^P I ■ '^\ ff ^ 



dx V dxl dy 



d'P I d\ ,,, ^ d'P /_:_.. rf\ .„,.. d'P 



dx' 





also Q= (sin y -^j /G^) ! 



Hence, if we call the values assumed by (sin y —J /"(») a"d (cos «/ —j /"(«) at the point 

 under consideration A and fi respectively, (14.) and (15) may be written thus: 



2^'af= BSa^-2ASxSy-BSy' (16), 



= J^ii^ + 2BSxSy-ASy' (17). 



Let ^x = ^s cos (p, Sy = Ss sin cj), then 



2 -— = B COS 2 (p - A s\n 2 (p, 



s 



= A COS 2 <p + B sin 2(p ; 



or 2 s4= C'cos (2^ + a) (I8), 



0= sin (2^ + a) (19), 



by putting A = B tan a. 



Equation (19) determines two values for 2 (p + a, and one of tliese will make ^'« positive, the 

 other negative; hence at the point in question there will be two branches curved in opposite senses, 

 one will be a minimum, the other a maximum. 



I have proved this proposition for simplicity's sake in the case of a double point, but the same 

 mode of investigation may be applied to that in which the increments of « all vanish up to any 

 given order ; this I proceed to do. 



It is not difficult to see that if the coefficients of the powers of Sx and Sy in the increments 

 Sx, Pn §"'-'» all vanish, then the value of S^sf may be written thus, 



1.2 mS'"ss={lx— + ^y—] P (20), 



\ dx dyj 



