504 Mr. O'BRIEN'S CONTRIBUTIONS TOWARDS A SYSTEM 



33. If u be the formula of a right line (involving of course one variable parameter), the 

 formula of a plane containing that line is evidently 



ti + mv, 

 m being a numerical variable parameter, and v any determinate line symbol. 

 If the plane be also restricted to contain a given point u, its formula is 



u + m (u — ?«), or m u + m'u , 

 where m + m = 1 . 



34. Let the symbols of the angular points of a triangle be u, n , u" ; then the symbol of the 

 point mid-way between u and u" is 1 {u + u"), and the formula of the line drawn through 

 this point and u is 



(u + u 

 u + ni 



u\ , 



3m\ m , , „ 



1 - — M H (u + u + u ). 



Now if we put m = 1^, this formula becomes symmetrical with respect to u, u, tt':, which shews 

 that the point whose symbol is i {u + u' + u") is common to the three bisectors of the sides of a 

 triangle drawn from the opposite angles. 



35. We shall now give o few examples of this method applied to Mechanics. We have 

 already (in the Paper read a few months since) shewn how the fundamental principles of Statics 

 inay be proved and expi'essed with great simplicity by means of the symbol D. We have also 

 shewn bow the motion of a rigid body about its centre of gravity may be investigated by means 

 of this notation, and exemplified its use in the problem of Precession and Nutation. 



36. We may investigate the equations for finding the motion of a planet in the plane of its 

 orbit, and the motion of that plane, as follows. 



Let u be the symbol of the position of the planet at any time t, then the symbol of the force 

 acting on the planet will be 



Let r be the radius vector of the planet, a, /3, 7, three direction units at right angles to each 

 other, a being the direction unit of m (and .•. u - ra), and -y being perpendicular to the plane of 

 the orbit : let tu, denote the angular velocity of ji and y about a, wg that of y and a about fi, 

 (0,1 that of a and /3 about y ; then w., is the angular velocity of the planet in its orbit, ai, is the 

 angular velocity of the plane of the orbit about the radius vector, and w-i is evidently zero. Hence, 

 (see Equations 38, former Paper,) we have 



da d(i dy 



dt at at 



da f//3 dy 



Now u = ra\ wherefore differentiating and substituting for — , — , and — we have, 



dt dt dt 



du dr da 



^- = -;- a + r — - 

 at dt dt 



dr 



= -rr" +»'"'3P; 

 dt 



d'u d'r dr da ^(rajs) d/3 



■'■ rf?" " d? " "^ d< Tt'^ dt ^ ^^'"'li 



