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a and /3 independent of one another; then the equation to 
the surface which differs insensibly from that represented by 
u = 0, will be 
du . du 
u + ——. 6a + *777. 5/3 + &c. = 0 ; 
da d[i 
and the co-ordinates of the points in which these surfaces 
intersect must satisfy their two equations, that is, they must 
satisfy 
u = 0, 
du 
da 
.5 
a + 
du 
d/3’ 
¿/3 
+ &c. = 0 ; 
or if the surfaces be consecutive 
u = 0, 
du du 
+ X • Ti" ~ ~ 0, 
da ¿/3 
where X = limit of 5/3-f-5a; but since X may have any value, 
the latter equation resolves itself into 
du 
-r = °> 
da 
du 
d/3 
= 0 
these then together with u = 0, are the three equations for 
determining the curve of intersection of two consecutive sur 
faces ; from whence we may obtain its two equations involving 
only one of the parameters; and finally by eliminating that 
parameter we may obtain the equation to the surface generated 
by the perpetual intersection of the first series of surfaces.