290 
PEOFESSOE W. J. MACQUOEN EANKINE ON THE 
Hence ^observing that we have for an indefinitely large cylinder, 
2 kax 
' #=-+2. 
dr r ' 
and for an indefinitely large sphere, 
2 kax 
( 54 ) 
(55) 
Wax 
$=x-%.-pr, 
d$ a? v -Wax 
dr r + ^ ’ yA > 
/ 
in each of which expressions 2 denotes the summation of terms belonging to the several 
pairs of foci, if there are more than one pair — each term containing its proper parameter, 
k or k 2 , and its proper double excen tricity, 2 a(=a' + a" when those two distances are 
unequal). 
Substituting cos 6 for -, the functions within brackets in the integrals of equations 
(52) and (53) are found to have the following values: — 
Cylinder : 
<p-J— ^=K cos2<? — 2 “ terms in pi &c.) (56) 
Sphere : 
dcp r Wa cos 2 0 1 n 
Qd?-3= r ( cos 4—3+2 - — p terms m - 6 &c.). . . . (57) 
The terms in and higher powers of * vanish, because of the indefinite increase of r. 
The terms in cos 2 Q—\ and cos 2 5 — ^ disappear from the integration. Hence the integral 
in equation (52) vanishes altogether ; and that in equation (53) has for its value 
^jj* ^ . 2rr 2 sin Qdb= — ~2k 2 a; 
(58) 
so that we obtain finally, for the total energy of disturbance per unit of velocity and of 
density , if the disturbing solid is an indefinitely deep cylinder, 
e d=P>; (59) 
and if it is a solid of revolution, 
E„=!(D-f SHo). . 
(60) 
The ratio borne by the total energy of disturbance to the energy of the disturbing solid 
is 
for indefinitely deep cylinders, 
2E 
~T) 
-=l ; 
(59 a) 
