Motion of Solid and Fluid Bodies. 179 
we shall obtain, by integrating twice with regard to ¢, the following expressions, 
p=7,cosA, = 1.h,sind "= 7, cosA,, = I.k,sing 
qg=7,cosn,=I1.h,sing g =7,cosu, = 1.k,sind (42) 
r=r7,cosy,=I1.h,,sne h =7,cosvy,, =1.k,,sind 
Tr 
where we have I = ——; and 
2Qrr 
h, = a.lcosa + n.mcosB + M.ncosy 
+ a,(mcosy + ncosB) + a,(ncosa + lcosy) + a,(/eos8 + meosa ) 
h,, = B.mcosp + L.ncosy + n./cosa 
+ B,(mcosy + ncosB) + B,(ncosa + cosy) + ,(lcosB + mcosa ) 
h,,, = c.ncosy + m.lcosa + L.mcosB 
+ ¥,(mcosy + neosp) + ¥.(2cosa + lcosy) + v,(lcosB ++ mcosa) 
k, = a,lcosa + B,.mcosB + y,.ncosy 
+ L(mcosy + ncosB) + y,(ncosa + lcosy) + £.(lcosB + mecosa) 
k,, = a,.lcosa + B,.mcosp + v,.ncosy 
+ y,(meosy + ncosB) + m(ncosa + Icosy) + a,(/cos8 + mcosa) 
k,,, = a,.lcosa + B,.mcosB + ¥,.2cosy 
+ B,(mcosy + ncosp) + a,(ncosa + leosy) + n(lcosB + mcosa) 
We have, therefore, the following equations to determine the direction and mag- 
nitude of the two transversals : 
h,,cosv, — h,, cosu, = 0 k, cosv,, — k,, cosu,, = 0 
h,,cosX, — h,cosv, = 0 k,, cosd,, — k,cosv,, = 0 (43) 
hosp, — h,cosA, = 0 k,cosp,, — k,,cosA,, = 0 
r= LVM +H +h’, sing tT, = LVRP+h + sing 
If we suppose the six fixed ellipsoids (29) constructed, with any point in the 
plane of separation for common centre ; and make (p,,p,,—p,,. 77,» 7,,,) the per- 
