ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
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Before going further, I remark that, m being a positive integer greater than unity, 
we have 
z = P + Q ^□ = mx (mx 2 + y 2 )™ -1 + {mx 2 + y 2 + (m — 1) mx 2 } {mx 2 + y 2 'y n ~ 2 V□ + &c., 
the subsequent terms being divisible, the rational ones by □, and the irrational ones 
by □ VD. Hence, observing that 
mx 2 + y- + (m — 1) mx 2 = m 2 x 2 + y- = □, 
we see that Q contains the factor □, and the equation D=0 belongs to a cuspidal 
curve on the surface. If however m = 1, then the equation is z = x + \/\I\, so that 
Q = 1 does not contain the factor □ ; and □ = x 2 + y 2 = 0 is not a singular curve on 
the surface, but is in fact the reduced or proper envelope. 
The curve represented by the integral equation will pass through the origin 
(x = 0, y = 0) for the value 2 = 0 of the constant of integration. In fact, for this value, 
the integral equation becomes 
— y 2m {y 2 + (2m — 1) ¿r 2 }™ -1 = 0, 
which belongs to a set of 2m + {m — 1) + (m — 1) lines coinciding with the lines y = 0, 
y = ix^2m—\, and y= — ix^2m — 1 respectively. The directions at the origin are 
therefore p = 0, p = + i V 2m — 1, which are the same values of p as are obtained from 
the differential equation; viz. since this is satisfied identically at the point in question, 
proceeding to the derived equation, we have 
p (p 2 — 1) + 2mp = 0, 
that is 
p(p 2 + 2m— 1) =0; 
but it is to be observed that these values of p are different from the values given 
by the equation □ = m 2 x 2 + y 2 = 0, which are p — ± im. The reason is that the curve 
□ = 0 being, as was shown, a cuspidal curve on the surface, the equation □ = 0 is not 
a solution of the differential equation. 
If however m = 1, then the integral equation gives at the origin no longer three 
values of p, but only the value p = 0. The differential equation however gives, as in 
the general case, three values; viz. we have _p(p 2 +l) = 0; and the values j)=±i 
obtained from the factor p 2 +1 = 0 are precisely the values of p obtained from the 
equation □ = a? + y 2 = 0, which in the case now under consideration belongs to the 
reduced or proper envelope of the surface, and is therefore the singular solution of 
the differential equation. 
III. 
The two curves of curvature which pass through any given point of a surface are 
distinct curves, not branches of one indecomposable curve. In fact if P, Q are the two 
curves of curvature for a point A, then for a point A' on P the two curves of