1863.] of a Solid Body about a Fixed Point. fe§ 



and if p, q, r represent the compouents of the angular velocity resolved 

 about the axes fixed in the body, then, as is well known, the equations of 

 motion take the form 



a| +h| +g| =-F(r-r=) + (B-C)2.+Hri,-G;,j, ] 

 +H§ +b| +F J=-G(r=-/)-H2r+(C-A)rp+Fi>2, > (i) 



To obtain the two general integrals of this system : multiplying the equa- 

 tions (1) byj9, q, r, respectively adding and integrating, we have for the 

 first integral 



■- Ap' + Bq'' + C?'' + 2(Fqr-\-Grp + Upq)=h, .... (2) 

 where h is an arbitrary constant. Again, multiplying (1) by 



Ap+Uq+Gr, 

 Up + Bq+Fr, 

 Gj)+Fq +Cr, 



respectively adding and integrating, we have for the second integral 



{Ap + Hq + Gry + (H2) + Bq + Fry+(G2) + Fq + Cry=P, . (3) 



where is another arbitrary constant. This equation may, however, be 

 transformed into a more convenient form as follows : writing, as usual, 



<B=BC-F\ B=CA-G^ €=AB-11\ V= A H G 

 iF=GH-AF, ^=HF-BG, f^=FG-CH, H B F . (4) 



A+B + C=:S, GF C 



and bearing in mind the inverse system, viz 



VA=13^!I:-iF^ VB=ra-€^^ Vc=^B-f^^ 



we may transform (3) into the following form : — 



(AS-B-e)/ + 2(FS +if)^r 

 + (BS-e-^)2^ + 2(GS+^)rp ..... (6) 



+(cs-^-^y+2{m+^)pq=k\ 



which in virtue of (2) becomes 



(%-^)p' + (^-^)q' + {^-^y-^2(Stqr + l&rp+^pq)^/c'~-SL (7) 



This form of the integral is very closely allied with the inverse or reciprocal 

 form of the first integral (2), and is the one used below. 



In order to find the third integral, we must find two of the variables in 

 terms of the third by means of (2) and (7), and substitute in the corre- 



