1 



52 /. G. Barnard on the Gyroscope. 



sines of the angles made by tlie axes Ox^^ Oy^ and Oz ^ witli 

 the fixed axes Oz and Oy, 



These values are shown to be (vide Bartlett's Mech,, p. 172) 



cos x^ 0^=: — sin 6 siu cp cos X-^ Oy=.cos 6 cos ip sin 75 — sin ip cos (p 



cos y^ Oz^ — sin d cos 9 cos y ^ Oijzzzcos 6 cos ip cos g^-f-sin ^f sin qp 



cos z^ Ozzrz cos cos 2 J 0?/=sin 6 cos v^ 



The differential ang-ular motions, in the time dt about the 

 axes Ox^j Oy^^ Oz^^ will be Va:dt^ Vydt^ and i;;jC??. ^Ye may de- 

 termine the values of these motions by applying the laws of 

 composition of rotary motion to the rotations indicated by the 

 increments of the angles 0^ (p and i/k 



If 6 and go remain constant the increment dtp would indicate 

 that amount of angular motion about the axis Oz perpendicular 

 to the plane in which this angle is measured. In the same man- 

 ner d(p would indicate angular motion about the axis Oz^ ; while 

 dd indicates rotation about the line of nodes OxV. In using 

 these three angles therefore, we actually refer the rotation to the 

 three axes Oz^ Oz^^ ON^ of which onc^ Oz^ is fixed in space^ 

 another, Oz^^ is fixed in and moves with the body, and the tliirdy 

 ON^ is shifting in respect to both. 



The angular motion produced around the axes Ox^^ Oy^^ Oz^^ 

 by these simultaneous increments of the angles qr, 6 and v^, will 

 be equal to the sum of the products of these increments by the 

 cosines of the angles of these axes, respectively, with the lines 

 Oz, Oz^ and OK 



The axis of Oz^ for example makes the angles 0^ 0^ and 90^ 

 with these lines, hence the angular motion v^dt is equal (taking 

 the sum without regard to sign) to cos ddip+d(p. 



In the same manner (adding without regard to signs), 



Vxdt=QOB cCj O^tZ^+cos fpdO 

 and Vydt=cos y^ Ozdip-hcos (90°+<y) dO, 



But if we consider the motion about Oz^ indicated by dtp, posi- 

 tive, it is plain from the directions in which (p and v are laid off 

 on the figure, that the motion cos ddip will be in the reverse di- 

 rection and negative, and since cos 6 is positive d^f must be re- 

 garded as negative, hence 



Vxdt=d(p— cos ddip. 



The first term of the value of Vj^dt^ cos x, Ozd^^ [since cos a:^ Oz 

 (——sin 6 sin (p) is negative and dtp is to be taken with the 

 negative sign] is positive. But a study of the figure will show 

 that the rotation referred to the axis O.r,, indicated by the first 

 term of this value, is the reverse of that measured by a positive 

 increment of ^ in the second, and hence, (as cos (p is positive,) dO 

 must be considered negative. Making this change and sub^i- 

 tuting the values given of cos a:, Oz, cos y^ Oz, and for cos (90^^ 

 +9),— sin 9, we have the three equations 



