( 730 ) 



P. + P. + P3 + ^. = ^ 

 - P, + P, + P3 - P, = ^, 



P\ Pg + P3 P4 ==^ -^2 



P -i-P — P — P =.4. 

 Then we can write the rotationvector in the form : 



,a>3 i 4- ^<x>J ^ ^vj^k ^ h (jW, i + ^10 J -\- jtu, k) 

 or in the form 



fl {<Pl i + 7^2 i + ^3 ^O + ^2 (if'l ^" + ^^./ + V's ^O- 



The notations h and f are taken from the above-named dissertation 

 of Dr. W. A. WiJTHOFF; h is defined on page 67 ; f 1 and f 2 on page 78. 

 The moment of motion becomes 



(P, -i- P3) ,a>3 i + (P3 -r I\) .CO, j -f (P, + P,) ,o>, ^- + 

 + A |(P, + PJ ,w, i + (P, i- PJ ,a>,i + (P3 + P,) 3C0, k] 

 or in an other form 



1 6, |(Prr, + A, V^)^ + (P'^2 + ^2 »f'2)i + (P'/3 + ^3 ^'3)^1 + 



+ I f 2 l(Pi|'i + ^1^1)^+ (Pil'2 + ^^2 ^/ 2)i + (^l^ 4- ^3 r 3) ^'1- 



If fp and If? represent the rotation vectors in Rr and /?/, we can 

 write the rotation : 



f 1 'P + f 2 ^% 

 and the moment 



1 R {e, <p + f. If') + (è f,. (.1) V' + \ ^. (^4) f/), 

 where the notation {Ayp means : A^ (p^ i -[- A^ (pj + A^ (f^ k. 



The first and strongest of these terms falls along the rotationvector ; 

 for a body with equal squares of inertia it is the onlj one; the 

 second, which, together with the ^'s, becomes stronger as the body is 

 more asymmetric, we might call the "crossed moment" because its 

 right part is caused by the left part of the rotation and inversely. 



Let us put finally the moment of force in the form f 1 ft + f, r, 

 where ft and v are threedimensional vectors ; then the above given 

 formula of the vector can be broken up into the six following 

 components, given successively by the coefficients of e^ Ï ,8^i, e^ j , 



R'(p, -^ A,ip, = A, ijj, ^3 — .43 V'3 <P, + 2fXi 



R^i + A «^1 = A ^2 V'3 — -^3 ^3 n\ + 2^1 



Rip^ + ^2 ^i'2 = A ^3 <Pl — A ^h 'fz + 2ft, 



Pip, + ^, y, = .43 if, V'l — A <ri ^\ + 2r, 



R^, + A, V'3 = A V'l ^2 — A IP2 9 1 + 2fi3 



Ptp, -^ A,(f,= A, (f, ip, — A, (p, tpi + 2^3. 



