500 Mh. O'BRIEN'S CONTRIBUTIONS TOWARDS A SYSTEM 



14. The following is an example of the case where the variable parameter is a direction unit. 

 The formula of a sphere is 



u + re, 

 where e is the variable parameter, and r determinate. 



For u + re represents a point whose distance from the point u is indeterminate in direction but 

 determinate in length, being always equal to r. Therefore the formula u + re defines a sphere 

 whose centre is u and radius r. 



We shall now illustrate this method of Symbolical Geometry by the following propositions, 

 without attempting any systematic arrangement, as our only object is to shew the nature and use 

 of the method. 



15. To deduce the equation of the plane from the formula of the plane, namely, u + re + r'e. 

 Let w y z he the co-ordinates of the point represented by u + re + r'e', a^ y^ z^ of the point 



represented by u, let a /3 7 be the direction units of the three co-ordinate axes, and let 



e = aa + b^ + cy, e = a' a + 6'/3 + c'y. 

 Then we have 



va + y(i + xy = tt + re + r'e' 



= ,r a + y^/3 + z^y + r{aa + 6/3 + cy) + r {an + !>' fi + c'y) ; 

 and .-., equating coefficients of a, /3, y, w = x^ + ra + r a' , 



y = y+ rb + r'b', 

 z = ss^ + re + r'e, 

 whence eliminating r and r' we find an equation of the form 



Jw + By + C« = D. 



16. To express the formtda of the plane by means of the symbol D. 



If V be an indeterminate line symbol, and e a determinate direction unit, Dv . e denotes a line 

 of any length drawn at right angles to e in any direction. Hence it is evident that the formula 



re + Dv.e, or (r + Dv.)e, 

 represents the plane whose perpendicular distance from the origin is re. 



17. The formula of the right line drawn through the two points represented by u and u' is 

 evidently 



u + m(u' - u) 

 where m is a numerical variable parameter. 



18. Hence if u be the formula of any curve the formula of the tangent at the point u is 



u + mdu. 



19. To shew that the formtda of the osculating plane of the same curve, at the point n, is 



u + mdu + ncfu. 



Let PP' and P'P" be two consecutive chords of the curve; produce PP' to any point Q, and 

 draw QQ' of any length parallel to PP" : then Q' is any point of the 

 plane containing the two chords, which plane, when the chords are 

 indefinitely small, becomes the osculating plane. 



Let u u u" be respectively the symbols of the points P P' P"; then 

 the symbols of PP' and PP' are respectively u — u and u" — u, and 

 therefore the symbols of PQ and QQ' are m{7i' — u) and n{u" - u), 

 m and n being arbitrary numbers. Hence the symbol of the point Q', 

 and therefore the formula of the plane containing the two chords, is 



u + m{u' — m) J- n(ji" — u). 



