XXXI. On a New Notation for expressing various Conditions and Equations 

 in Geometry, Mechanics, and Astronomy. By the Rev. ]M. O'Brien, late 

 Fellow of Cuius College, Professor of Natural Philosophy and Astronomy 

 in King's College, London. 



[Read November 23, 1846.] 



The notation Du . u, the meaning and use of which is explained in the following pages, denotes 

 a line of a certain length perpendicular to the lines denoted by the symbols u and «'. It is derived 

 from the consideration of the rotation of a rigid body, in which the line ?< is fixed, about the line u , 

 being, in fact, the differential coefficient of u with respect to the directions of the axes of co-ordi- 

 nates, the line n' being constant, as will be explained. 



It will be found, that this notation and a corresponding notation, A2<'.m, have several 

 important properties, that they express with great simplicity several conditions and equations in 

 various parts of Mathematics, and especially in Mechanics, and that they simplify in a remarkable 

 manner several complicated investigations. 



The present paper contains an explanation of the meaning of the notation, and its application to 

 Statics, and to the determination of the Rotation of a rigid body about its centre of gravity. 



unit of 



(Fig. 1 



Of the Notation D u'. ii. 



1. Let us assume the symbols a, f^, y to denote the lines OA, OB, OC, each a 

 length, drawn from an origin O at right angles to each other, so forming a 

 system of three rectangular axes. Let x, y, ~ denote any three abstract 

 numbers; then xa, yfi, zy will denote three lines, drawn along (or parallel 

 to) the three axes, and numerically equal to x, y, z respectively. 



Let OP be any line drawn from 0, and let us assume the symbol u to 

 denote OP in magnitude and direction; then, if xa, yfi, xy be the co-ordinates 

 of /-", we have, according to well-known principles, 



n = Xa + yfi + zy (1). 



We shall now suppose that the axes OA, OB, OC are capable of motion 

 a!)out the point O, always however remaining at right angles to each other. 

 We shall also suppose that ,r, y, z are not affected by this motion, or, in otiier words, that the 

 position of P relatively to OA, OB, OC, does not alter. In fact, we assume that the point P and 

 the axes OA, OB, OC are fixed in a rigid body which is capable of motion about the point O. 



Let S denote any indefinitely small displacement arising from a motion of tliis kind ; then from 

 (I) we have 



Su = xtiu + ySj3 + zSy ('.;)• 



Now, since a is invariable in length, ^a denotes a displacement of tiie point A at right angles to 

 OA : for, let OA' he the line drnoted by a + Su ; then, since 

 OA' = OA + A A', we have a -H ca = a + A A', and therefore 

 6n = AA'. Dut, since OA' = OA (a being invariable in length), 

 and since tlie angle O is indefinitely small, AA' is perpendicular to 

 OA. Hence oa denotes a dis])laeement of A at right angles to OA. 

 Vol. VIII. Part IV. 3 II 



(Fig. 2.) 



