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



7- If fJ. be any numerical quantity, we have 



D/ji u'. u = - Du . fiu' = - ixDti . u' = nDu'. u. 

 Hence we have 



DixU .u = iJ.Du'. u (' I)- 



From which it appears that a numerical coefficient of ?«' may always be brought outside the 

 sign D. 



Hence Dii . u = D {x'a + i/'/3 + ^'7) . u, 



= D.v'a . u + -Dy'/3 . u + Dx'-^ . u, by (8) ; 



and therefore by (11) Dit'. u = x'Da . u + j/'-D/3 . m + z'Dy .it (12)- 



8. In the equation ((>) putting all the co-ordinates, except ,r' and y, equal to zero, we find 

 Dm' a . y/3 = x'yy, and .'. Da . j3 = y: and in the same way we may shew that Dj3 . y = a, and 

 Dy .a = li. We have therefore 



Da.(i = y, DjS.y^a, Dy . a = (3 (13). 



From these equations we find by (7), 



Dfi.a = -y, Dy.f3=-a, Z>a . 7 = - /3 (14). 



Also we evidently have, 



Du.u = (15). 



And therefore 



Z>a.a = 0, Z)^.(3 = 0, X>7.7 = (16). 



9. Die . u is a line proportional to, and drawn in the same direction as the small displacement 

 iiu, which displacement takes place on the supposition that li is invariable : in other words, the dis- 

 placement iu results from giving a small angular motion, round the axis u\ to the rigid body in 

 which OA, OB, OC and Pare fixed. Fiom this consideration we may easily see that Du.u is at 

 right angles to ti! and ?«, and is proportional to )• sin Q*. 



It is plain from figure (.S), that the rotation by which the displacement Zu is generated is right- 

 ha?ided, supposing that we look along the axis of rotation (m') towards the origin. We may say, 

 therefore, that Du'. u is generated by riglit-handed rotation round the axis it'. 



10. Since Da . /3 = 7, and Da. 7 = - /3, it follows that {Da)'.ji = - /3 : and in the same 

 way we may shew that (Da)'. 7 = — 7 ; but, since Da . a = 0, we have {DaY. a = 0, instead of - a. 

 Hence {DaY written before /3 or 7 is equivalent to the sign - , and therefore Da . is equivalent 

 to the sign (-)-, or \/— 1 ; but this is not true of Da . written before a. Similar remarks may 

 be made respecting Dfi, and Dy. 



In general, we may see from what has been said above, that (^Dii)'-. u = — u when the numeri- 

 cal value of?/ is unity, and u' is perpendicular to u: in this case, therefore, Du'. is equivalent to 

 (-)i, or \/^. 



In this case, therefore, a line numerically equal to u, drawn at an angle 6 to u, and at right 

 angles to ii', is expressed by the formula 



u cos 9 + {Du. u) sin 9, or e 



SDi,-. 



U. 



11. When two ore more of the symbols Da., Dfi., Dy . come together, the order in 

 which they are written must not be changed : thus Dfi . Da . |3 = «» but Da . D(i . /3 = 0. 



• The ratio of D u'. u to r sin 6 is arbitrary ; we may therefore assume it to be r', and then we have D u'. u = r r' sin 0. This 

 is equivalent to the assumption that, \a = x', Xb=y', \c ~ z', in Article 4. 



