330] 
115 
[329 
fixed curve OP 
or OM'; then 
WN'S" is the 
is, SS" = 2SN; 
330. 
3us of S', is a 
of X, but of 
hing, if instead 
of twice the 
s epicycloid the 
d the point S. 
rays proceeding 
(or pedal) in 
ed rays. 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
[From the Philosophical Magazine, vol. xxvi. (1863), pp. 373—379 and 441—452.] 
I. 
Consider the integral equation 
Az 2 + 2Bz + G = 0, 
where z is the constant of integration: the derived equation is 
n = (A C' + A'C- 2BBJ-4 (AC-B*) (A'O' - B'% 
= (CA' - O'A) 2 - 4 (AB' - A'B) (BO' - B'O) , = 0 ; 
and if for greater simplicity we write A = 1, then the derived equation is 
O = C'*~ 4<BC'B' + 4CR 2 = 0, 
corresponding to the integral equation 
+ 2 Bz + (7=0. 
Writing the integral equation under the form 
(z + X) (z + Y) = 0, 
we have 
whence also 
2 B = X + Y, O =XY, 
2 B' = X'+ T, C = XT + X'Y, 
and the derived equation becomes 
0 = —(X— Yy X'Y'. 
15—2