IN GEOMETRY, MECHANICS, AND ASTRONOMY. 419 



Of the Notation A u'. u. 



12. In obtaining the notation Did ■ u we supposed the axes a, /3, 7 to be varied in position, 

 but not in length, always remaining at right angles to each other; we shall now obtain another 

 notation by supposing the axes to undergo a different kind of variation. 



Let S denote anj variation (whether in length or position) of the axes a, ^, -y, my« being sup- 

 posed invariable : then 



^M = .r^a + J/5/3 + sr^y. 

 Let us assume that 



la = x'lh, 5/3 = y'lh, Sy = x'Sh, 



where Sh is a small displacement in the direction of the line n', or ,v'a + »//3 + z'y. 



Thus we have 



Su = (.v.v'+ yy'+ zz')Sh. 



xa'+ yy' + xz is therefore the differential coefficient of ?<, when the axes a, /3, y suffei' the vari- 

 ations x'lh, y'lh, z'hh respectively, i.e. when the points A, B, C (fig 1.) receive displacements 

 proportional to x', y', z' respectively in the direction of the line u'. We may therefore represent 

 this differential coefficient by the notation A„7f, since the magnitude and direction of the variation 

 of u depends upon u, or is, so to speak, a function of ti. We have therefore 



A„.M = !i!ai'+ yy'+ zx'. 



It is evident from this expression that we may interchange u and u'. Also the operation 

 A„ is clearly distributive, and we shall therefore, as before, write Au'.u instead of A„-?«. Hence 

 we have, 



Ati- u = wx + yy + zx (17), 



or Am. J< = rr cos d (l^)- 



A?/, u = i^u .u' (19), 



and A (ti'+ u") . !< = A u'. m + A u" . u (20). 



13. The following formulae are also evident, namely, 



Am . M = 9-^ (21). 



If u' be at right angles to u, then 



Am'.?< = (22). 



Hence it follows that, whatever u' be, 



A?/. {Du'.u) = (23). 



14. We may express ,vyz and u by the following formulae, 



A a . M = *, A^. u = y. Ay .u = z (24), 



u = aAa . u + fiAfi . u + 7A7. u (25). 



(2.5) may be expressed by saying that 



«Aa + /3A/3 + 7A7 = l. 



