244 CELESTIAL MECHANICS. I. vii. 26. 



+ (Br sin (p—Aq cos (p) sin ^-—N' 

 {Br sin <p—Aq cos <p) cos %|/ — {Aq cos 9 sin ^ + 

 Br cos cos ^-H-Cp sin 6) sin >!'=: — JV"; 

 Nheing=SJ{Q:v'—Py')dtBm,N=zSJ{Rx—Pz') 

 dtDm, andN''=3f{Ry—Qz)dtj)m, (343); and 

 af^y\ and z being the coordinates, referred to 

 the centre of gravity, and parallel to J7, y, 

 and z. ' (C) 



We have first, for x\ y' and z', which are the ar, ^, and 

 £ of article 324, 



x'—x" (cos 5 sin 4^ sin 9 + cos rj. cos (p)-\-y" (cos fi sin t|/ cos 



^— cos >|/ sin (p)-\-z" sin fi sin i|.. 

 y''=:.x" (cos 5 cos %|/ sin cp — sin 4^ cos <p^-\-y" cos d cos \|/ cos 



^ + sin ^^ sin <p)+2" sin 6 cos %J/. 

 z'zzLz" cos fi — y sin fi cos ^ — x" sin 5 sin ^ : [and if we sub- 

 stitute for these equations, in order to shorten a ver}^ tedi- 

 ous reduction, a;'=zaa:''+%'' + 7^'',/=^^'' + fy" + ^2"; and 

 then, in order to obtain the value of x'di.y —y'dix' (343), 

 make d.x' zz a! x" -\- y" ^ y' z" , and d/zzSV + n'/ + ^;2", we 

 may omit in the products all the terms containing x"y", 

 x"z", ory'z'f since their sum vanishes for the whole body, 

 and we shall obtain a result in the ^ovmA'x"^ + B'y"^-\- Cz"\ 

 which may be transformed into ^{B'+C — A)A-\-^ {A' -{■ 

 C—Bf)B^-\{A'-\-Bf—C') C, for the whole body, since 

 J.=:S(/2_|.2'^2)j3m, B-S (x''2 + s''2)D^^ and C=S {x"^ + 

 y"^)Dm. Now for x'Ay—y'dix', we have a^x"- + ^Ey"^-\- 

 y^z"^-a!^x"^—0By"^—y'l:z"^, and A'z=.al'—a% B=^,'-> 

 0s, and C=y^-yr 



Again, 

 a=cos 9 sin ^|^ sin ^-fcos \^ cos (p 



