FORM OF TANGENTIAL EQUATION. 
3G9 
4. The coefficients in equation (4) are deserving of notice. Equated to zero, they 
are the tangential equations of curves which possess interesting geometrical relations to 
the curve U. For the purpose of examining their properties, let the curve U be given 
by the equation 
K, «i, « 3 • • • • • • 5 »X#> yY~ l 
+ n ' . . . 0,x®>y) ,, " s + &c - : =o ; ( 6 ) 
then substituting in this the value of ^ from equation (3), and equating the result with 
equation (4), we get the following system of identities : — 
A 0 =(ff 0 , a x , a 2 . . . — tf) n =0, 1 
A. l =vt{a l . a 2 . . . «„X1, ! 
A 2 =v 2 f(« 2 , « 3 , . . . <£1, “ t) n ~ 2 
+(<*, . . . <vXl, -«)”- ! =0, \ (?) 
A 3 =vH 3 (a 3 , a 4 . . . a„Xl, —t) n ~ 3 | 
+ 3*T(J 3 , i 4 .. .5J1, -^)”“ 3 
+ 3 vt(c 3 , c 4 . . . c„Xl, -O” -3 
(4, • • • 4X 1 * -0”" 3 =0> 
&c. &c. &c. j 
5. The system of identities (7) are remarkable for their symmetry, the equation 
A 0 =0 being independent of all but the coefficients of the highest powers of x and y, A, 
of all the homogeneous terms lower than the (n — l)th in x and y , &c. Transformed 
into the usual form of tangential coordinates, they become 
A 0 =(a 0 , 0„ «2 • • ■ <$>, — X)”=0, 1 
^=*( 0 ,, a 2 , a 3 . . . a n Jp,— a)”- 1 
—^(5,, b 2 , b 3 . . . ^Xj^ — k) n_I = 0, 
A 2 =v-(a 2 , a 3 . . . — \)” -2 
— 2p(M 8 .. ■ &-XK ^ 
+^ 2 (c 2 , c 3 . . . 5„X^, — x)"- 2 =0, 
A 3 = v\(l 3 , « 4 . . . ff n X», — ?0’ 1-3 
— 3p 2 (J a , b 4 . . . 5„X^, —x) 71 " 3 
+ 3 ^( c 3 , c 4 . . . -*)"“ 8 
^ • • . <y>> -A)"- 3 =0. 
3g 2 
