414 MISCELLANEOUS PROPOSITIONS. 



here A, B, C, D, E are certain differential coefficients of 

 <f> (x, y) ; and R may be symbolically expressed as 



1 ( , d d \ 



where v denotes <j> (x + th, y + tk), and t is some proper frac- 

 tion. 



Let us suppose that A and B are not both zero ; assume 

 A = Ksiny, and jB = ^Tcos7; also put r cos 6 for h and 

 r sin for k. Then the equation (f> (x + h, y + k) = becomes 



r ( 1 



-TTsin (7 + 0} + - \ C cos 2 + 2D sin 0cos0 + E sin 3 ^ 



" * ) 



+ 7 = (1). 



72 



It is obvious that when r is infinitesimal is also in- 



r 



finitesimal ; and that the above equation is satisfied by a 

 value of for which 7 + is infinitesimal, and by a value 

 of for which 7 + is infinitesimally different from TT ; 

 and by no other value of except such as differ from these by 

 a multiple of 2tr. Hence we have an ordinary point of the 

 curve. Therefore for a singular point it is necessary that 

 A = and JS = 0. 



Suppose then that A = and B = 0. The equation (1) 

 reduces to 



tan 2 0+tan0 + +J = (2). 



.0 Jli) r 



385. Suppose that D 2 is greater than CE\ then we know 



22) n 



that tan 2 6 + -~- tan d + -^ can be resolved into real factors ; 



MJ Mf 



and so may be expressed as (tan d tan a) (tan 6 tan /3) : 

 and a and ft may be supposed to lie between and TT. Thus 

 the equation becomes 



27? 

 p = (3). 



