of Edinburgh , Session 1867-68. 
225 
and equal to pt , and uv parallel to PT, the whole acceleration Vu 
may be resolved into Vv and vu; and P vu is an isosceles triangle, 
whose base angles are each equal to the angle of the spiral. Hence 
Vv and vu bear constant ratios to P u, or to SP or PT. 
The acceleration, therefore, is composed of a central attractive 
part proportional to the distance, and a tangential retarding part 
proportional to the velocity. 
And, if the resolved part of P’s motion parallel to any line in 
the plane of the spiral be considered, it is obvious that in it also 
the acceleration will consist of two parts— one directed towards a 
point in the line (the projection of the pole of the spiral), and 
proportional to the distance from it, the other proportional to the 
velocity, but retarding the motion. 
Hence a particle which, unresisted, would have a simple har- 
monic motion, has, when subject to resistance proportional to its 
velocity, a motion represented by the resolved part of the spiral 
motion just described. 
If a be the angle of the spiral, w the angular velocity of SP, we 
have evidently 
PT . sin a = SP . (o. 
Hence 
Vv = P u = pt — y = - 0> PT = . ° J , SP = n 2 . SP (suppose) 
1 SP sin a sin- a y 
and vu = 2Pv . cos a = - wC0S a PT =k . PT (suppose.) 
sin a 
or 
sin 2 a ’ 
Thus the central force at unit distance is n 2 = — , and the co* 
2w cos a 
efficient of resistance is k — 
sin a 
The time of oscillation is evidently — ; but, if there had been 
a> 
no resistance, the properties of simple harmonic motion show that 
2 ? t 
it would have been — ; so that it is increased by the resistance in 
oi ' * 
the ratio cosec a : 
1, or n : 
4* 
2 o 
VOL. VI. 
