318 Dr. Hirst on Equally Attracting Bodies. 



the general integral of which is easily found to be 



m^j^= — log [c(mcosCT+ sinT!7)12 + CT; . (21) 



or, since 



cosOT= — =^ = e'"^ 



> . . . . 21« 



■=\/l 



J 



<^=flog[c(mf+V^lI^)]Vcos-f. (22) 



In this equation, which represents the one or the other group 

 of equally attracting curves, according as the upper or lower signs 

 are taken, c is an arbitrary constant, p is retained as an abbre- 

 viation for the function of 6 given in (20), and represents of 

 course the variable perpendicular from the pole upon the tangent, 

 which is the same for all curves, and 



0<COS *- <77. 



r 2 

 By writing the differential equations fulfilled by all the curves 

 (22) in the forms 



d0~-pV r^' 



dp " p'^\ r^' 



where the upper and lower signs correspond throughout, we 

 easily recognize that the curves which constitute the group to 

 which the upper signs in (22) belong are, like the given spiral, 

 always concave towards the pole, whilst those of the group cor- 

 responding to the lower signs are, on the contrary, always convex. 

 In order to arrive at a clearer conception of the nature of these 

 curves, however, we may remark generally that the equation 

 (17), or 



r= -^—, (17) 



cos OT ' 



when 7s is constant, is that of a curve of the same nature as the 

 limiting curve (15). Hence, considering ot as a variable para- 



TT 



meter which may assume all values between and -^, (17) repre- 



sents a system of curves of which the limiting curve and that at 

 infinity are the initial and final individuals, corresponding re- 



