176 Dr. Hirst on Equally Attracting Bodies. 



value of y\r in (16), contains. The ai'bitrary constant a indi- 

 cates, as above remax-ked, that a surface may be turned in any 

 manner about the a?-axis without ceasing to belong to the same 

 group of equally attracting sm-faces. 



20. Between every pair of equally attracting surfaces, how- 

 ever, a general relation exists which leads easily to the determi- 

 nation of as many such pairs as we please. With a view of find- 

 ing this relation, we observe that the equation (11) of art. 13 

 may be written thus : 



fdu duA/du duA 1 /du duA/du ^^i\_(\ 



\dd'^IdJ\dd~dd)'^'^i^[d4>'^d^)(d^~d^)~^' 



or, setting for brevity, 



2v=u + Uj,\ 



thus : 



d0' dO'^sm^ddcji d(f> ^ ' 



But if p and p, be radii vectores of new surfaces, such that 



z;=log^, 



where <y, 7, are arbitrary lines, it is evident, on the one hand, 

 from (17), that 



7 V e c, 

 7, V c ?•, V c c', ' 



that is to say, that p is proportional to the geometrical mean 

 between r and r,, and p, to the geometrical mean between ?• and 

 r'l, the inverse of ?•,; whilst on the other hand, it appears from 

 the equations (18) and (10) that these new surfaces [p) and (p,) 

 have their corresponding normal vector-planes perpendicular to 

 each other. We conclude, then, that if two surfaces (r) and (r J 

 attract the pole in the same manner, and (r',) be a third surface 

 inverse to either of the former, the two surfaces p= -v/r.rj and 

 p — \/r.i\, whose radii vectores are respectively the mean pro- 

 portionals between r and v^, and between r and v\, will have their 

 corresponding normal vector-planes perpendicular to each other. 



21. The converse of this theorem is also true, and will be 

 useful to us; that is to say, if [p) and {p■^) be any two surfaces 

 whose corresponding normal vector-planes are perpendicular to each 



