. 426 



Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS 



for this represents a line drawn in the same direction as m'- u, and having its numerical magnitude 

 equal to 



where R is the numerical magnitude of ti — ti [observing that A («'- u) . (ti'- u) = R" (Art. 13, 

 equation 21)]. 



Now A (?<- m) . (jt — m) = All. II — 2Au . ii'+ Am . ti 



: r' - 2 Am . u + r^ (Arts. 12 and 13). 



Therefore, since - is very small, we have 



i» , ■,( Am.mS 

 5 A (m - m) . (2« - m) } t = r ' I 1 + 3 ;^— I very nearly. 



j (m'-m); 



2Z)m . USm = — 7^ Du'. (luom + — 2m Am . u'Sm). 



ml A M . M 

 Hence CJ = — 1 +3 7-— 



r H '■ 



and therefore, since Du . u = 0, and Du'.u = - Du . u', 



m 



r' 



Now "EuSni = 0, since the origin is centre of gravity : also, by Art. 12, equation (17), we have, 

 observing the properties of principal axes, 



luAu .u'Sm = 2(a?a + y/3 + zy) (xx'+ yy'+ xz')Sm 



= a^'aliW'Sm + y'^'^y^Sm + z'y'S.z-Sm 



= u'S.r'Sm - (Ax'a + By'j3 + Cx'y), 



since 2a'-^»i = 2/'5m - 2(j/°+ s;^)om, &c. &c. 



. 3 m' , 



Hence, since Du'. u'= 0, we have 2I>m . Ubm = —^Du . {Ax a + By fi + Cx'y)*. 



Thus (42), cleared of the sign 2, becomes, 



— (Aw,a + fia),/3 + C(U37) = -^Z)?/. (^x'a + By'^ + t'^'7) ... (44). 



29. To find the Solar Precession and Nutation by means of this equation. 



Let 7 be the north polar axis of the Earth ; then B = A, and C exceeds A by a small quantity, 

 X^ suppose, and therefore C = A{\ +\). Hence, observing that (o,a + w^ji + 0137 = <«, ,va + j/'/3 

 + ^7 = u, and Du> .w =0, Du. u= 0, (44) becomes 



dtti ^ d{m,y) 3m , 



dt dt r" ' 



(45). 



In the parts of this equation multiplied by the small quantity X, we shall suppose that the Earth 

 revolves about its polar axis with a uniform angular velocity, and that the Earth moves round the 



• Performing the operation Du'iK.e. x Da . + y' D^. + x'D~,.) 

 the second member of (43) becomes 



3m' 



— {{B- Oy'zcL + {C-A)zx'fi+{A- B)x'yy]. 



The coefficients of o, p, y here are the well-known expressions for 

 the moments of the attraction of Sim or .Moon about the principal 

 axes of the Earth. i 



