553] 
TWO SMITHS PRIZE DISSERTATIONS. 
563 
of curvature on the consecutive surface. In the orthogonal system they must be so; 
and this imposes upon the infinitesimal normal distance p, a condition; viz. it is found 
that p considered as a function of (x, y, z) must satisfy a certain partial differential 
equation of the second order. 
It hence appears that no one of the three families of surfaces can be assumed 
at pleasure; for taking the equation of a family to be p — f(x, y, z) = 0, then p being 
the value of the parameter for the given surface of the foregoing investigation, and 
p + Bp the value of the parameter for the consecutive surface, the normal distance at 
the point (x, y, z) between the two surfaces is 
viz. Bp is here a constant ; and we have 
satisfying the foregoing partial differential equation ; or, what is the same thing, p 
considered as a function of x, y, z must satisfy a certain partial differential equation 
of the third order; viz. this is the condition to be satisfied in order that a family of 
surfaces p —f{x, y, z) = 0 may belong to an orthogonal system. 
71—2