Johnston — Initial Motion. 5 



for these equations express that there is no motion of the point Xi y^ Si 

 in the direction k »h ^h, &c. &c. Taking as axes of reference, axes 

 fixed in space which coincide with the principal axes through the 

 centre of inertia, we have from these equations and equations (5) 

 and (6), 



i*i + Bi{l+ \u) + ^2 (cos 012 + \i2) + Sz (cos an + A13) "I 



+ Ei (cos ai4+ Xu) +-ffo (cos 015+^15) = 



P2 + Bl (cos ai2 + \12) + i?2 (1 + A22) + -R3 (cos a23 + A.23) 



+ Ei (cos 024 + A24) + i?5 (cos a25 + A25) = 



P3 + El (cos 013 + Ais) + -S2 (cos 023 + ^23) + -S3 (1 + ^33.) ' 



+ Ei (cos 034 + A.34) + Eo (cos aso + A35) = ' 



Fi + El (cos au + Mi) + E2 (cos au + Mi) + i?3 (cos asi + A34) 



+ Ei[l+ A44) + Eo (cos 043 + A45) = 



P5 + El (cos 015 + Ais) + E2 (cos a25 + A25) + E3 (cos 035 + A35) 



+ Ei (cos 045 + A45) + -So (1 + A55) = ^ 



where 



/ z 3/ m 



P^liX+ mi Y+ niZ + m [pi— + q\ —+ ri - \ , 



(8) 



All 







A12 ■ 





012 s the angle between tlie lines hniini, ^2»«2«2. 

 &c. &e. &c. 



Therefore we can at once find the values of the initial reactions 

 .Ri, Ro, . . . R-o by solving these equations. It is obvious that when 

 there are only four constraints the equations for the reactions are the 

 first four equations with the last term of each omitted. When there 

 are only three constraints the first three, with the last two terms of 

 each omitted, and so on. 



"When there is only one constraint the initial reaction is given by 

 the equation 



Pi + Pi (1 + Au) = 0. 



This case and several of the subsequent ones, when the system of 

 applied forces reduces to a single force through the centre of inertia, 

 have been investigated by the late Professor Townsend^, and he has 

 given interesting geometrical constructions for the initial reactions. 



Quaiteily Journal of Mathematics, vol. xiii., p. 284. 



