ANSWERS. 229 



(12) L-^saA 



(14) The equation = p?x 2 K gives : (a) when K Z > /x 2 , 

 dr dt 



x = e~ Kt ( C 1 X' t1 -'*" + ^-^-^O, from which it can be shown that 

 the particle approaches the centre asymptotically, reaching it only in 

 an infinite time ; if C l and C 2 have opposite signs, the particle will 

 do so after first reaching a maximum elongation and then return- 

 ing. (&) When K 2 = /x 2 , x = e~ Kt (C l + C z t). (c) When K 2 </x 2 , x = 

 Ci<e~ Kt sin(V/x 2 K 2 /+C 2 ), and the particle performs oscillations of 

 period 27T/V/A 2 * 2 and of decreasing amplitude C^""'. 



Pages 36, 37. 



(1) i watt = 0-73737 foot-pounds per second = 0-001 341 H.P., 

 i H.P. = 745 -9 watts. 



(2) i metric H.P. = 735-75 watts = 0-9863 British H.P. 



(3) 27! H.P. (6) nearly 200 gallons. 



(4) 49i H - p - (7) 35*54 gallons. 



(5) (a) 64; (J) 224384. (8) i hour. 

 (9) () 88 hours; (c) about 21 weeks. 



Pages 46, 47. 



(2) Taking the axis of z vertically upwards, U= U Q mg(z %) ; 

 the equipotential surfaces are the horizontal planes z = const. ; the 

 potential energy is V = mg(z ZQ) . 



(4) Taking the fixed line as axis of z, U\f(r)dr\ the equi- 

 potential surfaces are circular cylinders about the axis of z. 



(5). Let r = V(x-x o y + (y-y o y + (z-z l) ) 2 be the distance of 

 the moving point (x, y, z) from the fixed centre (x , y , z ) ; then the 

 direction cosines of the central force P=f(r) are 



= =, ==, 



r dx r dy r dz 



where the + sign holds for a repulsion, the sign for an attraction ; 

 hence X= f(r) , F= /(r) Z= f(r) |", or, putting 



X=dF(r)/dx, Y=dF(r)/dy, Z= 

 and finally U= F(r). 



