170 



Dr. Hirst on Equally Attracting Bodies. 



Substituting these values iu equation (2), we find at once the 

 following analytical expression, in polar coordinates, for the tan- 

 gent of the acute angle, A=A|r, between the radius vector and 

 normal at the point M, 



From this expression we deduce, too, 



, 1 



cosy= 



x/ 



1 + y +sm 



i^^V 



d<j>J 



Again, by equation (4), the direction cosines of the normal 

 with respect to the rectangular axes MO, MB, MB' are, respect- 



^^®v> , du , du cos ylr 



cosyfr, -^cos^r _ - . _^-. 

 ^ du ^ d(f) sm d 



If M, be the corresponding point to M of any other surface («,), 



the direction cosines of the corresponding normal M,Nj, with 



reference to the rectangular axes M]0, MjB,, M,B', respectively 



parallel to MO, MB, MB', will be 



, du, . du, cos "dr. 



cos^„ ^.cos^„ ^-^, 



where yfr^ is the acute angle between the normal MjNj and the 



same radius vector MO. Consequently, if we repi'esent by NNj 

 the inclination of the corresponding normals to each other, or, 

 more strictly, the angle enclosed by two intersecting lines parallel 

 to these normals, we have 



A . , \ -, du du, , 1 du du, \ ._,, 



cosNN,=cost.costH^l+^'^+^i^^-^-^7- (7) 



Again, by the fundamental formula of spherical trigonometry, 



cosNNi— cosyjr .cosi/tj 

 sin yjr . sin i|rj 



is the general expi'ession for the cosine of the angle v between 

 the corresponding normal vector-planes ; the equivalent expres- 

 sion in polar coordinates therefore is, by equation (7), 



'du du^ I 1 du du^~\ 



cos V = cot ^ . cot "^x-s-Ja 



or more fully by equation (6), 

 du dui 



+ 



-F?i- ^ + 



dd ^ sin^ e d(f> d<l>J' 

 1 du du^ 



dO dd sixx^d d(j> d<^ 



/{duY 1 {duy / {duiV, JL^{^V 

 V \dd) "^ sin^0\d<j>) V \d0j ^ sm^6\dcf>) 



.(8) 



