/. G. Barnard on the Gyroscope. 







51 



> 



^ \ 



the resultant of components about other axes (which may always 

 be reduced to three rectangular ones) : and by this means, about 

 whatever axis the body, at the instant we consider, may be 

 revolving, its actual velocity and axis are determined by a 

 knowledge of its components t^x, Vy? ^t, about the principal 

 axes Ox^ Oy^ Oz^^ these components being, as with lineal ve- 

 locities, equal to the resultant velocity multiplied by the cosine 

 of the angles their several rectangular axes make with the re- 

 sultant axis. 



As the true axis and rotary velocity may continually vary, so 

 the components t;^, t^, v^^ in equations (1) are variable functions 



of the time. 



Fig. 1. 



^1 



For the purpose of determining the axes Ox^^ Oy^ and O2,, 



Ox. 



iliary angles are used. 



If we suppose the moving plane of .7;, j^,, at the instant con- 

 sidered, to intersect the fixed plane of icy in the line AW and 

 call the angle xON=\p^ and the angle between the planes jry 

 and x^y^ (or the angle 2 6fe,)-i9, and the angle NOx^~(p^ (in 

 the fi";urc, these three angles are supposed acute at the instant 

 taken^ these three angles will determine the positions of the axes 

 Ox.. Ou.. Oz.. Cand hence of the body") at anv instant, and will 



themselves be functions of the time; and the rotary velocities 

 ■^x, Vy? ^z'^ may be expressed in terms of them and of their dif* 

 ferential co-efl^cients. 



For this purpose, and for use hereafter in our analysis, it is 

 necessarv to know the values, in terms of g», ^ and M', of the co- 



