380 Mr. Hennessy on the Connexion between the Rotation of 



hemisphere, and ^j, ^ ^3, &c. the moments of inertia of the 

 zones of the southern hemisphere, ^j and ^j being those of. 

 the zones in the immediate vicinity of the equator. Then J 

 the moment of^jn^if^ of the entire shell will be obtained by 

 the equation ^(0\ 



the external shell being so thin that the variati<5^ in its den- 

 sity may be neglected, and its moment of inertia found as if 

 it' were homogeneous. Let the zone ^j be included between 

 the latitude vj/j and the equator, the zone ^2 between rj/i and 

 t^2) and so on to the zone 0„, included between ^n-i and v^^, 



TT 



or — • Similarly, let ^„ ^c^ and i!,^^, be included respectively 



between the equator and dj, 6 and ^2, 9^_i and 9,„. 



The values of the moments of inertia of the zones will be 

 then found by the equations which follow, where A^ Bj, Ag, 

 B2, &c. represent the semi-polar and semi-equatorial axes of 

 the imaginary spheroidal shells which correspond to each 

 zone of the northern hemisphere; and A'^, B'l, A'2, B'g have 

 a^ similar meaning with respect to the zones of the sou^therh 

 Hemisphere. ifi aiiJna aiu Ju J .ericq^iJi to^aivxii^ 10 ^anJ 



^i=:f^' JpAiB^cos^^rf^ 





^' ., 'Mil.) *^ V'l * . 1 • 



Jff^iarf'ddt' * .n^93b sH) I0 





M12.) 



^,„=/*^^pA'„B'Vos^r/^ 



llie values of Aj, Bj, StcT; A'^, B\, &c., mi^st be found fpi; 

 each zone by the following method. ; 



Let T represent the thickness in feet or miles of a zone of 

 the external shell, the mean latitude of the zone, « the polar 

 semi-axis of the earth, /3 its equatorial semi-axis, D the depth 

 of the lower surface of the shell below the surface of the sea, 

 R^ the radius of the earth at 0, and R the radius of the in- 

 ternal spheroid at the same latitude. Then 



R = R^ - (D - T), Rgz=^/ (aS sin^ O + ^^ ^052 e). 

 Let y correspond to «, )j to /3, as D — T, or G does to Ro. 



^Ln -^- « (R^ ± G) - , ^ (R^ ± G) 



