OF SYMBOLICAL GEOMETRY AND MECHANICS. 501 



Now in the limit we may put 



u — M = dw, ?/" — 11 = du + d-%1. 

 Hence the formula of the osculating plane is of the form 



u + mdn + nd^u, 

 putting m in place of m + n. 



20. From \\\\s, formula to deduce the ordinary equation of the osculating plane. 

 Let X y ss he the co-ordinates of P, then 



u = xa + yfi + zy, 

 and therefore the formula of the osculating plane is 



(x + mdx + nd^x)a + (y + mdy + iid'y)fi + (a + mdx + nd?z)y, 

 whence, if x^ y^ z^ be the co-ordinates of any point of the osculating plane, we find 



x^ = X + mdx + nd^x, y, = y + mdy + nd^y, ss^ = z + mdz + nd z, 

 from these equations, eliminating the variable parameters m and n, we find the common equation 

 of the osculating plane. 



21. Respecting the geometrical meaning of the symbols dti and d'u it is worth observing, 

 that du represents in magnitude and direction the element (ds) of the arc of 

 the curve defined by the formula u, and d^u represents what is called a 

 double sngitta, as we may prove very easily ; for, let P P" P" be three conse- 

 cutive points of the curve indefinitely near each other ; complete the parallel- 

 ogram PP' and draw the diagonal P' Q. Let ?< ?/ id' be the symbols of the 

 points P P P', then tt" - u' represents the line P' P'\ and therefore the line 

 PQ,, and ?/ — u represents the line PP' ; hence, it follows, that the line P'Q 

 is represented in magnitude and direction by (m" — u) — {u — u), or, passing 

 to differentials, by d'U. P'Q is a double sagitta of the arc PP" *. 



22. From this we may derive the following remarkable theorem. 



If ii be the formula of any curve in space, « the numerical length of the arc of the curve 



measured from any fixed point to the point u, v the numerical magnitude of d'u, and e the 



ds- 

 direction unit of d'u, so that d^u = ve; then 2 — expresses the numerical length of the chord of 



V 



curvature drawn in the direction e. 



The direction e is perfectly arbitrary, depending on what the independent variable in the 

 differential d'u is supposed to be. If we consider s to be the independent variable, it is evident 

 that PP' and PP" are equal in magnitude, and therefore the chord of curvature becomes the 

 diameter of curvature. 



23. Another remarkable theorem is the following : 



The symbol d ( — ) represents a line drawn from the point of contact towards the centre of 



curvature, and numerically equal to the angle of contingence. 



This may be proved as follows, du represents a line whose length is ds drawn in the direction 



of the tangent at the point u, therefore -— represents the direction unit of the tangent. Hence, 



ds 



if we draw two direction units from the same 'point parallel to two consecutive tangents, the symbol 



• If we take « as the indepcntlcnt variable, in which case rP'= P'P", P'Q will he perpenilicular to PP", and iP u will reprcHenl 

 the double augitta pointinj^ tuwards the centre of curvature. 



