[ erf or Differentials. 191 



4. Quantities such as SVA- involve operating on v by both 

 V and d. These operators are not always commutative. In 

 fact if P be any scalar, and cr and r any vectors, whose dif- 

 ferentials we may call (f>c/p and Odp, we shall have 



VStVP = StV . VP-6>'VP, by (5) of the first paper, 

 = SrV . VP + i&diVP +/SfyVP + &S0&VP, 



and this extended to a vector by the usual method gives 



V<£t= -St\7 . S/cr + i<f>di+j$6j + k$Qk. . (10) 



This equation may be obtained in a quite different way. 

 Write 



d<f)T = d(p . t + fair, 



where d(f> . r indicates the result of differentiating <f>T as if r 

 were a constant vector. With this understand in «• 



dxfyj = SdpV . StV • or f 4>&dp 

 ='StV . SdpV * <t -f (f>0dp, 



provided we do not substitute for dp any but constant 

 vectors. If now we call the two terms on the right <f> x dp and 

 fytflp* we shall obtain from each a part of \/(pr. The first 

 term gives 



q x = /StV . Sz'V . a +JStV • S/7 • o- + £StV . S£ V . a- 

 = StV . (zS/'V . a- +jfS;V - * + &S£V . <j) 

 = -SrV.Vcr 7 

 and the second term gives 



q 2 = i(p0i+j<f>6j+k(f>0k, 



leading to the same result as before. 



5. From (10), by putting p for r and ^ for <£, 



¥X y= 'd l V v + X X* x + f l X*f* + v X**'- • . (11) 



Here the first term on the right is the same as -fv— — 9 ; and 

 because for anv direction at right angles to y 



X 2 -w 2 x-f m,-0 [12) 



