50 /• G. Barnard on the Gyroscope. 



The analysis I sliall presentj so far as determining the equations 

 of motions is concerned,* is mainly derived from the "vvorks of 

 Poisson (vide ''Journal de TEcole Polytech." vol. xvi — Traite de 

 Mecaniqiie, vol. ii, p, 162), Following his steps and arriving 

 at his analytical results, I propose to develop fully their mean- 

 ing, and to show that they are expressions not merely of a visi- 

 ble phenomenon, but that they contain within themselves the 

 sole clue to its explanation: while they dispel all that is myste- 

 ^ rious or paradoxical, and, in reducing it to merely a "particular 

 case" of the laws of *' rotary motion," throw much light upon 

 the significance and working of those laws. 



Although not unfamiliar to mathematicians, it may not be 

 uninteresting to those who have not time to go through the long 

 preliminary study necessary to enable them to take up with 

 Poisson this special investigation ; or whose studies in mechan- 

 ics have led them no farther than to the general equations of 

 *' rotary motion" found in text books, to show how the particu- 

 lar equations of the gj^roscopic motion may be deduced. 



In so doing I shall closely follow him ; making however some 

 few modifications for the sake of brevity and of avoiding the 

 use of numerous auxiliary quantities not necessary to the limited 

 scope of this investigation. 



The general equations of rotary motion are (see Prof. Bart- 

 lett's "Analytical Mechanics" Equations (228), p. 170): 



In the above expressions the rotating body (of any shape) 

 A B CD (fig. 1) is supposed retained by the fixed point (within 

 or without its mass) 0. Ox^ Oy and Oz are the three co-ordi- 

 nate axes, fi^ed in space, to which the motion of the body is re- 

 ferred. Ox^^ Oy^^ Oz^j are the three principal axes belonging 

 to the point 0, and which, of course, partake of the body's 

 motion. The position of the body at any instant of time is 

 determined by those of the moving axes, 



Aj 5 and C express the several "moments of inertia" of the 

 mass with reference, respectively, to the three principal axes 

 Ox^ Oy ^ Oz^\ Ny^ i/, and L^ are the moments of the accelerat- 

 inri forces^ and v^-^Vy^Vx^ the components of rotary velocity^ all 

 taken with reference to these same axes. 



Like lineal velocities, velocities of rotation may be decomposed 



that is, a rotation about anv sinerle axis mav be considered as 



' www 



r(i.) 



I 



