k 



/. G, Barnard on the Gyroscope, 63 



from one plane to another, as if no rotation existed; and (as a 

 corollary) as little expenditure of work. 



But deflecting forces are devclopedj by angular motion given 

 to tlie axis, and normal to its direction, which, are very sensible, 

 and are mistaken for direct resistances. If the extremity of the 

 axis of rotation were confined in a vertical circular groove, in 

 which it could move without friction ; or if any similar fixed re- 

 sistance, as a material vertical plane, were opposed to the c?e- 

 fleding force, the rotating disk would vibrate in the vertical 

 plane, as if no rotation existed. Its eq^uation of motion would 



d become that of the compound pendulum, V7 = ^|-X^^- What 



then is the resistance to a change of plane of rotation so often 

 alluded to and described? A misnomer entirely. 



The above may be otherwise established. If in equations (3) 

 we introduce in the second member an indeterminate horizontal 

 force, g\ applied to the centre of gravity, parallel to the fixed 

 axis of y, and contrary to the direction in which, in our figure, 

 we suppose the angle V to increase, the projections of this force 

 on the axes Ox^^ Oj/^j will be a'^' and b^g^ and the last two of 

 these equations will become, (calling cosines x ^ Oy and y ^ Oy^ 



^ a' and h\) 



Adva:-\-( C— A)nvydi=. —7 M{bg^h 



Multiplying the first by Vy and the second by v^^ and adding 



A {vy d Vy -j-v^c dvx )z=LY M\g {avy^hv^) dt-\-g' (a' Vy ^b'va;)d {]. 



But {avy—hva:)dt has been shown (p. 63) to be =c?.cos<9, — and 



(a'Vy'-h'Vx)dt 



simil 

 d. (sin Q cos ^). (For values of a' and 5', see p. 62.) 

 Let us suppose now that the force g^ is such that the axis of 

 the disk may be always maintained in the plane of its initial po- 

 sition xz. The angle V' would always be 90"^, ^V'^O, and d. (sin^ 

 cos«/')=0. That is, the co-efficient of the new force g' becomes 



zero : 



A{vy^+v^^)=2rMg 



as before (p 



But the value of Vy^+v^^ likewise reduces (since -^=0) to — 



^ ^ dt ^ dt^ 



and the above becomes the equation of the compound pendulum, 



{g) T^— "~S~^^ ^^^ Q-^-h^z— (cos 5— cos a), {h being determined.) 



announced 



:o prevent any dejledion 



s to cause the axis to vil 

 if no axial rotation existed. 



be 



