2 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



complete and so entirely satisfies the conditions expressed in our quotations above from that 

 work, as to leave nothing to be desired. But it does not appear to me that his method, 

 which depends essentially on the summation of the centrifugal forces, is so widely applicable 

 beyond the limits of this particular problem as that by which the same results are obtained 

 in this paper: but be this as it may, any new point of view, if, a true one ("vrai point de 

 vue") has its special advantages, and on this ground may claim some attention. 



SECTION I. 



The Method, with some kinematical Applications. 



1. As we shall here be concerned only with the directions of lines in space, and not with 

 their absolute positions, it will be convenient to suppose them all to pass through a common 

 origin O, and to define the inclination of two lines as OP, OQhy the arc PQ of the great 

 circle, in which the plane POQ meets a sphere whose centre is O and radius constant. We 

 shall also suppose any linear velocity, acceleration or force, represented by a length along OP, 

 to tend from O towards P, and any angular velocity or the like, represented in like manner, 

 to tend in such a direction about OP that, if OP were directed to the north pole, the direction 

 of rotation would coincide with that of the diurnal motion of the heavens. 



2. Let u denote any magnitude, which can be completely represented by a certain length 

 along the line OU, and which can be combined with a similar magnitude v along OF by means 

 of a parallelogram, like the parallelogram of forces or velocities. Then of course u may be 

 resolved in different directions by the same principles, and thus if we adopt rectangular 

 resolution, the resolved part of u along OP will be mcos UP, which may be denoted by Up. 

 We proceed to inquire how Up varies by a change in the position of OP. 



3. Suppose OP to be a line moving in any manner about O, and that it shifts from OP 

 to a consecutive position OP' in the time dt ; and conceive that this motion arises from an 

 angular velocity Q about an instantaneous axis 01. Resolve Q into its components Qcos lU 

 about OU and Qsin/i7 about a line in the plane lOU, perpendicular to OU: and farther 

 resolve this latter component in the plane perpendicular to OU into the components Q sin lU 

 . cos lUP in the plane POU and Q sin lU . sin lUP perpendicular to the same plane. 



Then the component in OU and that perpendicular to it in the plane POU produce 

 displacements of P perpendicular to the arc UP, and consequently do not ultimately alter the 

 length of the arc UP, so that Up remains ultimately unchanged so far as the motion of OP is 

 due to these components : but the component perpendicular to the plane POU increases UP 

 by the arc Q sin lU . sin lUP . dt, and therefore the increment of Up from this component (being 

 equal to - u . sin UP . d . UP) is 



- mQ sin UP . sin lU . sin lUP . dt. 

 But the other increments being zero, this is the total increment of Up, wherefore we have 



^^= - mQ sin lU. sin f7P. sin IUP...{A). 

 dt 



