ACTED ON BY NO EXTERNAL FORCES. 777 



attains its maximum or minimum value ; the equations being linear between (cos a)*, 

 (cos ^y, (cos yY, say between x, y, z, the extreme values of z of course correspond to the 

 zero values of x and y respectively. 



In using such tables of treble entry, we may suppose the initial angular velocities about 

 the principal axes to be given, from which and the known moments of inertia the quan- 

 tities L and M may be calculated, and then by the direct or supplemental method of reduc- 

 tion the value of X and of the two parameters g",, q^ in the equivalent disk, each less 

 than unity, found. 1st. If the reduction is direct — from the given inclination of the axis 

 of the disk to the invariable line — the time t„ from the epoch can be found by inspec- 

 tion, and then the entries corresponding to t-\-to will give the inclinations at the end of 

 the time t of the principal axes to the invariable line, and the position of the node 

 defined as the intersection of the invariable plane with the plane through the invariable 

 line and the axis of the disk (which axis coincides with a known one of the two extreme 

 axes of the given body), and also the total angular velocity ; the corresponding position 

 of the node and value of the total angular velocity of the original body are then known 

 by simple arithmetical computations from the theorems above given, involving K only for 

 the first, and X, L, M for the second. 2nd. If the reduction is contrary or supplemental, 

 we have only to substitute the supplemental angles of inclination to the invariable line 

 in determining t,,, and proceed in all other respects as before, taking the supplements of 



Hence 



M , dz 



where 



and 



V qj \2, ?i + 2a/ 



q^+q,\/z,zj\ q.+qj {i-z-)^z,z^ 



The limiting values of z correspond to Z,=0, Z,=0, or, which is the same thing, to the values of z when y and x 

 are successively made zero in the equation (1). 



It may he useful to the reader to he enabled to compare the above values of t and ? in terms of z with the 

 equivalent determination of Legendee, Exerc. du Cal. Integ. tome ii. p. 334, 



VIZ. idt= _ 



, _ 2tan^ /w+1 fZ(^ d^ \ 



^"l-msin/J \^ -/l -(^(sin 0~ {\ + (tan ^)»(8in ^y) V 1 -c>(8in «^)V ' 



for this purpose it will bo necessary to bear in mind that Legeitdre's A, B, C are not the momenta of inertia 

 themselves, but the elements out of whose binary combinations they are formed, and that his middle magnitude 

 is not B but A ; the reader wiU then find it necessary to trace the values of Legendre's i, W, t, <J/, 5, /3, m, n, ^, c 

 by the formulae and definitions given at pages 334, 319, 328, 315, 321, 322, 325, 319 6m, 333, and possibly some 

 other which has disappeared from my notes of the Exercises, tome ii. 



