Dr. Hirst on Equally Attracting Bodies. 283 



42). When « = 0, b^=l, this equation (41) takes the simpler 

 form 



{c—x)x [c + x)x 



and, with reference to the generation of this simpler surface, the 

 circular cubics (0), (0,) become equal circles around the origin 

 as centre with radii equal to c, their planes become perpendicu- 

 lar to each other, and bisect the angles between the planes of 

 the circles (C) and [C^, which latter have now radii equal to each 



other and to ^. 



46. When i = and a=l, the equation (40) assumes the 

 still simpler form 



and represents the same right cylinder already obtained, by a 

 quite different method, in art. 33, 



47. When tan i/r has the form " ^ „ , where /(^) is any given 



function of alone, the fundamental equation (6) of Art. 34 

 may be treated in a manner precisely similar to that followed in 

 the 35th and following articles. Instead, however, of extending 

 the present memoir to greater length by entering into further 

 details, we will only remark, in conclusion, that it will be some- 

 times found more convenient to treat this equation (6) in the 



ordinary manner, by setting P= -jn autl q= -^r, by means of 



which it becomes 



P'+^=^^^^^=t\ (42) 



where Hs a given function of 6 and <^. 



This being an identity, we may differentiate, separately, accord- 

 ing to 6 and ^, and thus obtain the equations 



dp q dq_ cos 6 ci_.dt 

 ^dd'^Biix'e'W ^^^ ~dB' 



^P ^ Q (^ ^A±^ 



d(}} sin^ dcf) d^' 



By means of the general relation 



dp _ dq 



d^~dd' 



and the origiual equation (42), the two last assume the forms 



