372 ON THE BIOTION OF 



The eight arbitrary constants introduced by the integration 

 of these equations are to be determined from the known val- 

 ues which the variables a", b", £,, n, and their velocities are 

 supposed to have at some given epoch of time. These eight 

 arbitraries are not the only ones of which the body is suscep- 

 tible. There will be ten in all, two being introduced by the 

 equation W= o, whose integral is 



CR-{-Fa"—Gb" = rf + e>, 



the constants s and e' being functions of the values which 

 a ", b", i?, and their velocities have at any given epoch. 



Let us now suppose that the distorsive moments of inertia 

 F and G are very small, in which case the rotation round the 

 normal may be increased to any assignable rapidity without 

 disturbing by that circumstance alone the smallness of the 

 oscillatory excursions. The equation W =■ o will now be 

 found reduced to Cdr = o, whence r = a constant quantity, 

 and R = rt-+- R\ R' being the angular distance of the first 

 body-axis from the first space-axis when / = o. Equations 

 (35) become at the same time 



dl = ( li—ry !l ^-^(rb" — da"), 

 dyj t = dYi,-\-rZ, — Z t \ra"-\-db"); 



d=k = d%—rdyi,, 

 d\, = d\~\-rdi, 



four linear equations which, in conjunction with the four 

 equations of motion transformed by the substitution of the 

 present values of p, q and r, will make up eight equations of 

 the first degree (six being of the secondhand two of the first 

 order) with constant coefficients. The^ equations may be 

 completely integrated either by D'Alembert's method, by 

 which we should be brought to twelve equations of the first 

 order ; or by eliminating the indefinite integrals £„ J?,', £,, >?„ 

 \m\ then proceeding by the method of exponential substitu- 



