20 



Sven Wickwell 



— = D " _JL = D" _L_ - D " 

 dU' 1 8F" J ' d W"~ 3 



(37) s a « 

 Then clearly 



A" = YnA + Y 18 Z) 2 + T 13 Z) 3 



(38) Z) 2 ' = Til A + 7 „ Z) 2 + t, 3 Z) 3 



A>" = Tai D l + Ï32 ^2 + Ï33 A . 



and using the symbols 



(39) rT . d TJ7 , = ZU D^ 1 ', 



v y a^a^gif'- 123- 



according to a well known theorem 

 (40) 



A" j A,"''- = (Tnfl. + YrA + Y.A)'' (ï.iA + Y 22 Z> 2 +Y 23 Z) 3 V (ï 31 Z> t +Y 32 Z) 2 +Y 33 Z) 3 ) ft , 



where after the development of the right membrum, we introduce the derivatives 

 by the equation (39). 



Now we immediately find the equation 



'i(U, V, W) 4- S B ijk D 2 j D s k = f"{U", V", W") + £ Bfa D^ H Z) 2 " J Z) 3 " Ä 



(41) X />V;; ; = 



= S Zi;;, ( Tll Z> 2 + Tt3 Z> 8 )< (t„ ZV+Ym Z) 2 + y 23 Z),) ' (Y 31 A+Tm Z) 2 +t 3 3 Z) 3 )* . 



Equating here the coefficients of Z) 1 ' / /)/ Z> 3 Y in both membra the ex- 

 pression of B rj o y{ in terms of B" jk [i-\-j-\-k = a-f ß+Y) is obtained. 

 Introducing the symbolical expressions 



(42) ^-^^(iP.^ 



where by (a, ß, 7) we mean the coefficient of a" &ß cY in the development of 

 (a + & 4- e) a +?+Y, 



we generally have 



(43) Z%* = (*, J, A) ( Tu a t + ï 21 a a +Y 31 y £ (T„ ^+Y 32 €,+Y m (Yi3 ^i+Y 23 ^ + Y 33 

 when after the development we again introduce the characteristics B'âfa by (42). 



Equations (36) and (43) give in a condensed form the whole theory of finding 

 the characteristics when the system of coordinates is affected by a rotation. 



