508 The Rev. R. Carmichael on Methods 
tained; the latter bemg the curve required, the former the pro- 
perty of the curve indicated by the given equation. 
2. Again, if it be proposed to integrate the differential equation 
(xda + ydy + 2dz)? =m? { (ydy—zdy)?+ (zdu—adz)*+ (wdy—ydz)’}, 
and at the same time to determine its geometrical interpretations, 
both immediate and remote, the same method can be employed 
with advantage. The equation obviously represents a curve.of 
double curvature ; and, remembering the expressions for the pro- 
jections on the coor dinate planes of “the infinitely small triangle 
standing on the element of a curve with its vertex at the origin, 
we have at once for the immediate geometrical interpretation of 
the given equation 
Q=mP, 
Q and P having the same meanings as before. 
For the remote geometrical interpretation, or the integration 
of the given equation, we have 
rdr=myr*dd, 
whence 
roe e 
where ¢ is the whole angle swept out by the radius vector. 
3. An inquiry obviously suggested by the last example, is the 
integration of the partial differential equation resembling it in 
form, ah 
(oY (tats ot Ba tee 
and the aes mination of its imetrical interpretation, if it have 
any. 
The equation obviously represents some surface; and if both 
sides of the equation be divided by 
du\? | (du\? | (du\? 
zz) + (Gq) +(e)» 
it will be easily seen that, yr being the angle between the radius 
vector to the point (z, y, z) and the perpendicular on the tangent 
plane at that point, the given equation is equivalent to 
r? sin? ap=Q*=m?7P3, 
or the given equation represents the surface for which the per- 
pendicular from the origm on any tangent plane is in a constant 
ratio to the intercept between the point of contact and the foot 
of the perpendicular. 
Now putting 
Q?=r?—P2, 
