OF DEFLECTIVE FORCES. 123 



ds being constant, the radius of curvature is 



^"" V(d^:t'- +cl^i/^ +d^z2)* 



We may first suppose a plane to pass through the tan- 

 gent of the curve and the ordinate z, which we may call 

 vertical, while x and y are horizontal, and a second plane 

 to pass through the tangent, and to be perpendicular to the 

 first or supposed vertical plane: then if the curve be pro- 

 jected vertically on this plane, the horizontal ordinates z 

 and y will be the same for the projection as for the curve, 

 and the tangent will be the same, and the curves will only 

 differ in length as the tangent diff*ers from the curve, the 

 elements of all three being ultimately in the ratio of equa- 



lity: the sagitta of curvature, 77-, of the projection, will 



be horizontal, being perpendicular to the first plane, which 

 is vertical; it is, therefore, the projection of the primitive 

 sagitta on a plane parallel to that of x and y : and the same 

 may be shown of two other projections on the other two 

 orthogonal planes, of x and z, and ofy andz, substituting 

 y and x successively in the place of z. 



Now, the sagittae of the three projections are -^ ^/{a^x* 

 -HAY), i ^/{A'^x^ + A^z% andi VC^V + a2z2), and the sum 

 of their squares is i VCa-^^ + a^ + aV), consequently the 

 primitive sagitta of which they are the projections, is i 

 {a^x^ + A^y^ + A^z% and the true radius of the curve 



AS^ _^ d£2 



^{a^x^ + aY + a^z'')" s/ (d2^2 + d Y + dV)* 



Scholium. The characteristic a is here substituted for 

 d in speaking of the sagitta, A being intended to represent 

 an actual evanescent variation, while the fluxion d is a finite 

 magnitude proportional to it (229). The student will 



