OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 545 
curve will be an irrelevant solution, unless it has the singularity in question on one or 
other of the two sheets. A cuspidal curve may yield us the true solution. 
Now it is easily proved that the osculating plane of a curve traced on a surface 
will coincide with the tangent plane to the surface when it touches one of the inflexional 
tangents, and only then. The osculating planes of the curve we are discussing will, 
therefore, be the tangent planes to the surface at the points where they osculate the 
curve. 
Thus the singularity under discussion consists of a cuspidal curve, such that the 
osculating plane at every point of it touches the surface at that point. The corre- 
sponding singularity on the reciprocal surface is of the same kind. 
§ 37. To prove this write the equation to the surface (multiplied by a factor, it may 
be) in the form \¢? = pw’, d, pw, ¢, w being functions of a, y, z, such that 6= y= 0 
are the equations to the cuspidal curve. 
Put 
= py =t, 
then 
b= Op, b= rp. 
Suppose now that 
o=2-+rhtoet. 
Par Uy ae 5 ar eae 
¢, and w, being homogeneous in 2, y, x and of the degree r. 
Then we may deduce expansions for y and z in ascending powers of ¢t and was 
follows— 
y=oa?+PBP+... 
+a (y+...) 
staves (Olle pres) es 
z=aeP+tPBet... 
+a(y?+...) 
+e (Fe? 4+...) +a7(C4...), 
the first power of t being absent throughout. 
More generally, we have in the neighbourhood of a cuspidal curve (t= 0) expansions 
of the form 
“x=e%,+ta,+Pa,+... 
Y=Ytimtlyt... 
2=%+%+02,4+... 
w=wt+tutPwt... 
MDCCCXCV.—A. 4A 
