The general Characteristics of the frequencyfunction of stellar movements 19 



number of motions having components within the intervals U" + x hdJ]"\ V" ± 1 l2dV"; 

 W"± 1 /-2iiW" are expressed through 



F"(U'\ V", W")dU" dV'dW", 

 and the number having components within the intervals U± l kdTJ\ V± l /*dV; 

 W± 72 d W through 



F{U, V, W) dUdVdW, 



and we have 



(34) V" = Tn U+-i,. 2 V+^W 

 ^" = Y 8 t ^+Y 82 r+Yss ^ 



As the Jacobiana of this transformation is equal to 1 we have 

 dUdVdW = dU" dV" dW" 



and accordingly 



F( U, V, W) = F (U", V", W") 

 when in F" we substitute the expressions (34). 

 Writing now 



V 8*«+* to" 



(35) F'dT", F", W") = ? "(U'\ V , W ) + L /i",, ^4 8r >/ 8 ^ 



(35*) /" = A" U"- + 7i" F" 2 + C" PK" 2 + 2D" F" H 7 " + 2E"U"W" + 2F" t7" F" 

 F(U, V,W) will also take this form. 



By the substitution (34) the determinant A" is invariant; consequently 

 A" = A. 



Substituting (34) in (35*) we obtain* 



(36) 



I = A" lu * + B"y 21 2 + C"* (si * + 2D" hl y 31 + 275" Tll ÏM + 2F" Tll Tll 

 I = A" ll2 * + £" Ï22 2 + C"y 32 2 + 2£>" Ï22 Y 32 + 2F" Tl2 y 82 + 2F" Yl2 Y 22 

 I = ^" T ,3 2 + 5"y 23 + C"y 33 2 + 2D"y 23 Y 33 + 2E"y 13 Y 33 + 2F" Yl8 T 23 



© = 4 "Yi2 Yis+ jB "Y 2 2 Ï23+ C "Y 32 T 8 b+^"(Ym Tbs+Tm Y 23 )+#"(t 32 Tu+Tsb Yi 2 )+^"(Yi 2 Y 23 +Yi 3 Y 22 ) 

 E = A "fu Yii4-- B "T 2 3 Ïii+C'ïbb »si +- D "(ïj8Ïsi + Yss Y 2 i) + #"(T 33 Tii+Tsi Yis) + F "('dz Yai + Tj t Y 23 ) 

 f - 4"Tu Yi 2 + B"y 2 i Y 22 + C'Ysi T 32 + Z>"(Y 21 Ï32 + T31 Y 22 ) + ^"(Y 3 , Yi2+Y 32 Tu) +^"(T a , T 22 + Ti2 Tsi) 



To transform the characteristics B" we introduce for a moment the 

 following symbols 



* Compare Charlier 1. c. p. 82. 



