172 Dr. Hirst on Equally Attracting Bodies. 



which we shall often find convenient on account of the greater 

 symmetry thereby introduced, these equations (10) and (11) 

 assume, respectively, the forms 



du du^ du dui _^ ,,„ v 



do) dco d(j> d(f) ' 



from which we easily deduce 



duj du 



d(o ~ d^' 

 duy_ — du 

 d<]> d(o' 



(12) 



:+5f.=o a3) 



where the upper and lower signs respectively correspond. On 

 differentiating the first of these expressions according to (f>, 

 and the second according to o), we ought of course to obtain 

 identical results; whence we conclude that, in order tliat the 

 two surfaces may possess the properties required, each must fulfil 

 the equation 



^ "' 



dw^ ' d<l>^ 



It is scarcely necessary to remark, that, in virtue of the last 

 article, if u and Uy possess the properties under consideration, so 

 also will their inverse surfaces u'— —u and u\-= — W) ; so that 

 if two of the four equally attracting surfaces u, u', Mj, m'j have 

 their corresponding normal vector-planes perpendicular to each 

 other, there will always be four distinct pairs which possess the 

 same property. On this account we need only consider one pair 

 of such surfaces — say that which corresponds to the lower signs 

 in (12). 



14. Conversely, if any surface (m) fulfils the equation (13), a 

 second, equally attracting surface (wj) may always be found whose 

 con'csponding normal vector-planes are perpendicular to those of 

 the former. For the hypotheses (12) are admissible when (13) is 

 fulfilled ; and Mj being determined accordingly, the equations 

 (10«) and (11a), which are perfectly represented by the system 

 (12), will of course be satisfied. 



15. The equation (13) plays an important part in physics; 

 its general integral is well known to be 



2u = ¥{co + i(f)) + F, {(o—i<f>), 

 where F and F, are symbols of arbitrary, real or imaginary func- 

 tions, and, as usual, i= v/ — 1 . By means of this and the equa- 

 tions (12), we find without difficulty 



2m, = F{(o + 20) — F, (&) — 20), 



