420 



Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS 



A7. i>a.= A/3. 



Aa.Z> 



7' 



...(26). 



1-5. Hence we may easily shew that 



Aa- (Dfi .It) = Ay .K, A^ . (Dy .u) = Aa . u, &c. &c., 



or, omitting ?<, 



Aa.Z?/3.= A7., A(i.Dy.= Aa., 



Afi. Da.=- Ay., Ay .Dji.=- Aa., 



Also [or from (23)] it follows, that 



Aa.Da.^0, A(i.D(i. = 0. Ay .Dy. = ...(^1). 



16. It is easy to see that the displacement which gives rise to the differential coefficient 

 An'. 11, is caused by a uniform e.vpansimi of the rigid body (in which the axes and the point P are 

 Hxed) in the direction of the line u, the modulus of expansion being proportional to the numerical 

 magnitude of u . That plane containing the origin which is perpendicular to u is unaffected by 

 this expansion. 



(Fig. 4.) 



Instances of the application of the Notation Dii. u and Au'. u to Statics. 

 17- The expression v, or 



xa + y/3 + ssy, 



determines completely the position of the point P ; on this account we 

 shall call u the symbol of the point P. 



In like manner, if X, Y, Z be the three components of any force, 

 and if 



U = Xa + Y(i + Zy, 



U is the symbolical expression for the force, representing it com- 

 pletely in magnitude and direction. We shall therefore call U the 

 symbol of the force whose components are X, V, and Z. 



For brevity we shall generally say, ^^ the force U" instead of, " ttie 

 force whose symbol is U ;"and, in like manner, " t/ie point u," instead 

 of, " t/ie point whose symbol is u." 



(!)• 



18. If the forces U, U', U", &c. keep a rigid body at rest, the six equations of eciuilibriuni 

 are contained in the following equations, viz. 



2^7= (28), 



2X>i«. U=0 (29). 



For 2fr=a2^ + /32F+ 7SZ, 

 and therefore (28) is equivalent to the three equations 



2X=o, 2r = 0, 2Z = 0. 

 Again, by equation (6) we have, 



Du.U= (Zy - Yz) a + {Xx - Zx) ^ + {Yx - Xy) 7, 

 and therefore (29) is equivalent to the three equations 



2 (Zy - Yz) =0, 2 (^Xz - Zx) =0, 2 (Yx - Xy) = 0. 



