140 The Rev. Samuel Haughton's Account of 



1. — Composition of notations. 



Let O be the intersection of two axes of rotation, OR, OR'; and let the 

 magnitudes of the rotations be represented by w, w'; then the motion impressed 

 upon the body by these two rotations will be the same as the motion produced 

 by a single rotation round an axis, which is represented in magnitude and po- 

 sition by the diagonal of the parallelogram formed by w, w. For, draw through 

 any point I of the body a plane perpendicidur to the line 01, and project upon 

 this plane tlie pai'allelogram formed by w, w'; — the sides of this parallelogram will 

 be o) sin ROI and u>' sin R'OI. Now the velocities impressed upon the point I 

 by the rotations w and w, are 01. id sin ROI, and 01. w' sin R'OI ; and the di- 

 rections of these velocities are perpendicular to the sides of the projected paral- 

 lelofrram. Hence, if this parallelogram be turned in its plane tiu'ough 90", its sides 

 will represent in magnitude and direction the actual velocities ; the resultant 

 of these velocities is perpendicular to the projection of the diagonal of the 

 parallelogram (w, m'); this projection, turned round through 90^ will represent 

 the actual velocity, which is tlierefore the same in magnitude and direction as 

 would be produced b}- a single rotation represented by the diagonal of {w, w). 

 Hence rotations may be resolved along three rectangular axes by the same laws 

 as couples, and they must be counted positive when the motion produced is from 

 z io X, X to y, y to z, and vice versa. 



II. — Linear Velocities i^roduced by a yiven Rotation. 



Let the origin of co-ordinates be assumed on the axis of rotation, and let the 

 magnitude of the rotation and of its components be represented by (w,p,q,r): 

 the velocity of any point (.r, y, z) is in a direction perpendicular to the plane 

 containing the axis of rotation and the point (,r, y, z) ; and its magnitude is re- 

 presented by the area of the triangle whose angles are situated at the origin, 

 the point {x,y, z), and the point {p, q, r). Hence, the components of the linear 

 velocity are represented by the projections of this triangle on the co-ordinate 

 planes. These projections are 



