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must satify these two equations; that is, they must satisfy 
» = v<p(a) + yf(a) + ^ (a), 
0 = oo\<p'(a) + &c.} + y \f (a) + &c.} + \J/' (a) + &c. ; 
or if the planes be consecutive, making ¿a = 0, the equations 
to their line of intersection are 
* - M(p{a) + yf(p) + ^ («) (0» 
0 = ¿v 0' (a) + yf (a) + y\/ (a) (2) ; 
and the general equation to developable surfaces will result 
from eliminating a between these equations; but as the 
elimination cannot be effected without fixing the forms of the 
functions (p, /, \Js, that is, without particularizing the surface, 
we must retain the above system to represent this family 
of surfaces, regarding a in the former as a function of cc and y 
to be determined by the latter. 
Cor. Hence the differential equation to developable 
surfaces can be obtained; for, differentiating (l) successively 
with respect to ce and y, regarding a as a function of co and y 
to be determined from (2), we have 
p = (p{a)+ {#<£'(«) +#/(«) +'/''(«)} 
<1 =/(«) + {•»$'(«) + yf i a ) +'K(«)} 
which by virtue of equation (2) are reduced to 
p = p(a), $=/(a); 
and since p and q are functions of the same quantity, they 
are functions of one another, 
••• P = *■(?), 
which is the differential equation of the first order to de 
velopable surfaces, containing one arbitrary function; this 
may be made to disappear by differentiating the equation