416 



Mr. OBRIEN, on A NEW NOTATION FOR VARIOUS EQUATIONS 



(Fig. 3.) 



,/ 



71, 



(3). 



In like manner ^/3 and ^7 denote displacements of B and C at right angles to OB and OC 

 respectively. 



2. Let the displacement Sa be resolved into two others, 

 Ac and Jb', of which Jc is parallel to OB, and Ab' to CO. 

 In like inanner let ^/3 be resolved into Ba parallel to OC, and 

 Be to ^O; and let ^7 be resolved into Cb parallel to OA, and 

 Ca to BO. 



Also let us denote the numerical magnitudes of these re- 

 solved displacements by c, b', a, c, b, a', respectively. 



Then, since OA, OB, OC always remain at right angles to 

 each other, it is evident that a = a, b= b', and c = c'. Hence, 

 giving these displacements their proper signs of direction, 

 namely ft, - 7, 7, - «, «> - /3, respectively, we have, 

 Sa = Ac + Ab'= ftc - yb\ 



Sft = Ba + Be = ya -ac > 



Sy=Cb + Ca = ab - fta.) 

 The quantities a, b, c here denote any arbitrary numerical differentials. 

 Making these substitutions in equation (2), we find, 



hi = (xb -yc)a+ (.vc - za) ft + (ya - xb)y (4). 



3. Now it is evident from the nature of the motion which § denotes, that Su represents an 

 indefinitely small line at right angles to t( ; therefore, if X be any numerical arbitrary quantity, \Su 

 will represent any line (not necessarily small) at right angles to 11. The sign \S therefore, written 

 before u, changes u into the symbol of a line at right angles to m, and therefore has somewhat the 

 same effect as the sign \/ - 1, or (-)■*. Since however there may be an infinite number of 

 different perpendiculars to u, it remains to put the sign XS in such a form as shall indicate 

 what particular perpendicular \hi represents. We shall do this in the following manner. 



4. Multiplying (4) by X, and putting Xa = .v', X6= y', \c = z, we find 



Xhi = {zy - z'y) a + (xx - x'z) ft + (y.v - y'x) y (5). 



Now it is evident from this expression, that Xhi vanishes when x = iv', y = y , z = z' ; in other 

 words, if we assume 



u' = x'a + y'ft + ^'7, 



it follows, that \^u = 0, when u = u. Therefore \^u denotes a differential* of u taken on the sup- 

 position that ti is invariable. 



On this account we shall replace X^ by the sign 2?„., defining £>,,. to denote a differential taken 

 on the supposition that ?/ is invariable. We have then, 



D^u = {zy- z'y) a + (<tz'- x'z) ft + {y ^v' - y'x) y. 



If we interchange ,r, y, z, and x', y , z' respectively, this equation becomes 



Dji'= (z'y - zy') a + (x'z - xz) ft + (y'x - yx')y. 



Hence we find, that 



i»„w'= - D,u. 



From this equation we may shew that the operation D„. is distributive with respect to u; that 

 is to say, that 



Meaning by tlie word differential here any quantity proportional to an indetinitely small difference. 



