S01.ID-; ON SIUI'ACKS. ( 1 ) 



By the substitution of these expressions in the second gen- 

 eral formula (1). it becomes integrable with respect to S: and 

 if we suppose the point O to be taken in the centre of gravity 



of the system, we shall have, after the obvious nil net ions. 



(14) 

 (MP&+SXJhnydS, -+- [ C + S(Zy — Vz )Dm}bP\ 



(Mr- K ,-hsriJ'ti)oK -+- [ /'-f-.v(A',c — z,x )i)„r\o(A = . 



(MPS, -+- SZ,Dm)6l -+- [ W+ S( V.r — X ;/ )Dm]bR ) 

 where 



(15) 



[ ' = Adp — Gdr — ffdq + (C — B)qr -+- F(f —q*) — Gpq -+- Hrp 



V = Bdq — Hdp — Fdr -+- (^ — C)rp +- Gfj> a — r m ) — /fyr + />, 

 W = Gdr — /Wy — Gdp -h(B — Jl)pq -+. Htf —p') — Frp + Gqr 



which are the same expressions as those which are given by 

 Lagrange in his first volume although obtained by a process 

 altogether different. 



These values would be greatly simplified by referring the 

 elements Dm to the principal axes of the body; but as the 

 axis which is vertical when a heavy body is at rest is not in 

 general a principal axis, it will be found accessary, in investi- 

 gating the phenomena of oscillatory motion, to retain the 

 terms multiplied by /'. G, If. quantities which may I think. 

 from their giving rise to a constant displacement of the instan- 

 taneous axis of rotation, he called with some propriety the 

 dutorriet moments of inertia. 



If the sy-tern is free, then by equating to nought the coef- 

 ficients of the six variations, we shall obtain six equations 

 determining the progressive and rotatory motion of the 

 body, namely. (1G) 



\ OL. in. — 1 s 



