1902 - 3 .] Isoclinal Lines of a Differential Liquation. 
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isoclinal family, the direction of the integral curve is the same as 
that of the isoclinal through P, the contiguous points Q and Q' are 
in general both on the same side of the isoclinal, but on opposite 
sides of the point P; therefore the sign of the variation of p 
changes on passing through P, i.e ., there is in general an inflexion 
on the integral curve. The condition for this is 
<l>x +P<f> y = 0 • 
This is equivalent to the usual condition for an inflexion ; for 
4> x dx + 4> y dy + cfipdp — 0 
but along an integral curve dy =pdx, hence 
djP_ = _ ( k±liy == o 
dx cf> p 
if <f> p 4= 0, which, along with 4= 0 
is the usual condition for an 
inflexion. Even if <j> p = 0 the geometrical reasoning shows there is 
still an inflexion, unless the ^-discriminant is an envelope locus for 
the isoclinal family. (See example (3).) 
This may obviously be generalised as follows : if the direction of 
the integral curve lie along the isoclinal at any point, and if the 
tangent at that point meet the isoclinal in n contiguous points, then 
it will also meet the integral curve in (n + 1) points. (See example 
( 6 ).) 
The following are a few examples in illustration of the above. 
Example (1) : 
p 2 -l- (3x + 2 y)p - \ x 2 + 3 xy + y 2 = 0 . 
