Dr. Hirst on Equally Attracting Bodies. 275 



effect that every surface of the class under consideration may be 

 turned in any manner around the a^-axis without ceasing to be 

 one of the equally attracting surfaces of the group to which it 

 belongs. In fact, if c be an ordinary constant, the value of the 

 quantity /(«) — oi.f'{u) in (29) will remain unaltered when /(a) is 

 replaced by /(«) + cu, and at the same time /'(a) will be changed 

 intoy(«) +c. So that if, for anj particular form oif{a), 



u='\lr[6, (f>) 



be the result of eliminating « from (29), 



u^ = -\Jr{6,<}> + c) 



will be the result of such elimination when /(«) is changed into 

 f{u) + CO.. But the equally attracting surfaces u and ii^ clearly 

 differ in position only, and both can be made to coincide by 

 turning either around the a?-axis through an angle c. 



37. We proceed to treat briefly a few particular cases of the 

 equation (29), art 35. If, in the first place, be a constant 

 =\, the equation (27) becomes 



W, «) = f^ i/X^sm^^-a^ ; 



whence, on integrating and subsequently differentiating accord- 

 ing to a, we find 



F(^,«) = -Xsm '(^^==j.-«tan '( ^^^^^ > 



da. \ a. cos d / 



so that the formulae (29) give 



^ . , / \COS 6 \ ., . rl, s • 



0=-tan M ] + (l^+f{u). 



^ acosa ■' 



(30) 



These equations, from which a is to be eliminated as soon as 

 / is particularized, represent the class of surfaces alluded to in 

 art. 8 ; each surface cuts every ray of a pencil, whose centre is 

 the origin, at a constant angle ^, whose cotangent is A.. 



According to art. 36, the hypotheses /(«)= A +Ba, and 

 /(«) = A = const, lead to essentially like results ; indeed, without 

 loss of generality, we may even sujjpose this constant A = 0, so 

 that the second of equations (30) becomes 



2 = *^^ 9> 



acosP 



T2 



