35I-] BODY WITH FIXED POINT. 



193 



Again, multiplying the equations (4) by T^, / 2 a> 2 , 7 3 a> 3 , and 

 adding, we find similarly, by (15), Art. 323, 



A V + A V + / 8 V=#* = const. (6) 



This is the principle of areas or of the invariable plane. 

 As, moreover, 



&) 1 2 + ft) 2 2 + ft) 3 2 = w 2 , (7) 



we have three equations (5), (6), (7) for determining a^ 2 , a> 2 2 , a> 3 2 . 

 Their solution gives, after some reductions, 



_ _ 



-- 



(A-AXA-A) (A-AXA-A) 



^ 



350. To find the time, multiply the equations (4) by 

 <w 2 /A> o) 3 // 3 , and add. This gives 



r 



^1 2 ( / 2 - / 3 )( / 3- / l)(A- / 2 ) 



= - M - - ' 



In this equation the values (8) should be substituted for j, o) 2> 

 < 3 . For the sake of brevity, let us put 



we then find 



- a> 2 ) (a> 2 - 7 2 ) 



This is an elliptical integral whose discussion is beyond the 

 scope of the present treatise. 



351. It remains to determine the position of the moving 

 system formed by the principal axes, with respect to a fixed 

 .system of axes through O, by means of Euler's angles 0, <, ty 

 (Art. 333). After finding co as a function of t from (9), we 

 have, by (8), co lf a> 2 , &> 3 as functions of t. Substituting these 



PART III 13 



