A “ smith’s PRIZE ” PAPER ; SOLUTIONS. 
485 
534] 
sin p is here an indefinitely small quantity of the order a 5 , all the terms are therefore 
at least of the order a-, and are to be neglected in comparison with a ; or neglecting 
such terms we have A — 0 (that is, the attraction of the solid NRQ is indefinitely 
small in regard to a); and the theorem is thus proved. 
10. Indicate in what manner the Lagrangian equations of motion 
d dT_dT_dV 
dt d% d% d% ’ 
lead to the equations 
for the motion of a solid body about a fixed point. 
The expression of the vis viva function T is 
T=l (Ap 2 + Bq- + Of 2 ), 
but this expression will not by itself lead to the equations of motion; we require to 
know also the expressions of p, q, r in terms of certain coordinates A, p, v, which 
determine the position of the body in regard to axes fixed in space, and of the 
differential coefficients A', p, v of these coordinates in regard to the time; each of 
the quantities p, q, r will be a linear function of A', p, v (p = aX' + bp + cv, &c.), 
containing in >any manner whatever the coordinates A, p, v. This being so, the equations 
of motion will be 
where 
only terms containing the differential coefficients of p, q, r, are the terms 
dp dp dq „ dq dr „ dr 
dX' dt + dX' ’ B dt + dX' ‘ C dt 
dp f 
dx’X 
there are of course two other equations only differing from this in that in place of 
A', they contain p and v respectively ; and since p, q, r regarded as functions of 
A', p, v are independent functions, the determinant formed with the differential coefficients