OF DEFLECTIVE FORCES. I21 



lies always in a plane perpendicular to the surface, and 

 the pressure of the surface could cause the motion to be 

 deflected in no other direction. That the curve must be 

 in a plane perpendicular to the surface, is evident from the 

 motion of all bodies rolling on each other, which come into 

 contact in directions at right angles to the surface : and 

 the wire must bend continually, from the position of the 

 tangent into that of the curve, in such a manner that all its 

 points descend perpendicularly upon the surface. 



267. Lemma A. The sum of the squares 

 of the projections of any right Hne on three 

 orthogonal planes is equal to twice the square 

 of the line. 



If the orthogonal ordinates of the line 5 be a, h, and c, 

 we have s^z=a^ +6^ -j-c^ ; now the projection on the plane 

 of a and b is VCa^ _j-&2)^ on that of a and c, >^{a^-\-C'); 

 and on that of 6 and c, »s/{h^ +c2) ; consequently the sum 

 of the squares of the three is a^ -\-b" +a^ -{-c^ -{-b^ -{-c^zz 



268. Lemma B. The fluxion of an arc is 

 as the radius of curvature and the fluxion of 

 the angular extent conjointly, or ds=rdd, and 

 the radius is ^= j^- 



The angle being measured in equal circles by the arc 

 subtending it, and in different circles being inversely as 

 the radius, when the arc is the same, it becomes evident 

 that the elementary arc of any curve must be as the angle 

 subtended by it, or as its angular extent and the radius of 

 the circle of curvature conjointly, the element of the curve 



