VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 5 



variation in direction of the acceleration of u are given for every instant. And we have also 



from the triangle FU'F' ultimately 



drj , da . 



— . sin/ui = -r7.smi/, 

 at at 



to determine — : and therefore we have equations to determine the intensity and variation in 



direction of u itself. Hence we have obtained a solution of the problem, " Given the path of 

 F and the variable intensity of/, to determine the path of U and the intensity of «," the whole 

 being referred to intrinsic elements. 



7. It will be useful to obtain results analogous to equation (C) for three rectangular 

 axes in a somewhat different form. Of course these might be obtained from that equation 

 itself, but it will be better to investigate them independently by the same kind of reasoning. 



Let u„ Uy, u^ denote the resolved parts of u along the moveable rectangular axes Ox, Oy 

 Ox, and let Q^, Q^, Q^ and /„ f^, / denote in like manner the resolved parts of Q and /. Now 

 by reason of the acceleration /, u^ receives in the time dt the increment f^ dt : also Ox changes 

 its position by reason of the rotations Qy, Q^, the first of which shifts it in the plane of xx 

 through the angle Qydt from 0^, and the latter in the plane of xy through the angle Q^dt 

 towards Oy] and from the first of these causes u^ receives the increment 



u^ cos I — \-Qydt\ + u^ cos (Qydt) - u^, 



or — u^Q dt ultimately, while from the second it receives the increment 



Uy COS ( - - Q^dt I + u^ cos (Q^dt) - u,, 



or UyQ^dt ultimately. Hence the total increment of u^, being the sum of these partial 

 increments, we obtain the equation 



du^ 



Similarly for Uy, u^ we should obtain 



du^ , \ {E). 



du 



8. To illustrate the applicability of these last obtained equations, we will select a few 

 particular kinematical problems. 



a. Relative velocities of a point in motion with respect to revolving axes. 



From the nature of the quantity u, it will be seen that it may be taken to denote the radius 

 vector OP of a point P, and u^, Uy, u, may then be replaced by the co-ordinates, w,y, z: also 

 /, denoting the acceleration of u, will in this case denote the absolute velocity of P, and f^ fy, 

 ft the absolute velocities resolved in the directions of the axes, which we will denote by v„ Vy, 

 v^. Then by the equations above we have three equations, of which the type is 



