2 The Astronomer Royal on the Conditions 



this to consider the resistance as a small quantity, for which an 

 approximation to the first order might suffice; and T then 

 undertook the investigation in its fullest generality, as regarded 

 the magnitude of the resisting force. 



In regard to the law of the resisting force, another difficulty 

 presented itself. It is known that, with large velocities, the re- 

 sistance varies nearly as the square of the velocity. This law 

 offers almost insuperable difficulties to mathematical treatment, 

 if we do not restrict ourselves closely by limitations as to the 

 form and position of the curve. But with the term depending 

 on the square of the velocity there is combined a term depending 

 on the simple power of the velocity ; and with movements so 

 slow as those of the cable in some parts, the term depending on 

 the simple power may perhaps be the predominant term. I 

 have therefore considered that no error of the least importance 

 would be introduced by assuming the resistance generally to 

 vary as the simple power of velocity, provided that the co- 

 efficient be so adjusted that the resistance at or near the 

 terminal falling velocity may be correct. As far as any error 

 is sensible it will be of this kind, that, for the lower velocities, I 

 have assumed a too great resistance ; the general effect of this, 

 however, will be neutralized by assuming the ship's velocity to be 

 a trifle greater than I have taken it in the numerical calculations. 

 There is another difficulty peculiar to the circumstances of a 

 moving cable. The cable moves longitudinally as well as 

 laterally, and there is a considerable longitudinal friction as 

 well as a lateral resistance. I have no data for determining the 

 coefficient of this friction. A small error, however, in the co- 

 efficient is not important, for the longitudinal motion of the 

 cable is always much less than its lateral motion. Under these 

 circumstances I have considered that the law and coefficient of 

 the longitudinal friction might, without practical error, be 

 assumed to be the same as those of the lateral friction. 



The combination of these two assumptions introduces great 

 comparative simplicity into the formulae ; for it makes the 

 friction in the directions of any rectangular coordinates exactly 

 proportional to the resolved velocities in the directions of the 

 respective coordinates. The differential equations resulting 

 from the mechanical considerations are therefore linear, and to 

 the facilities for solution peculiar to linear equations is entirely 

 due the success of the investigation. This inquiry, general in 

 its geometrical suppositions, but not wholly supported in its 

 physical assumptions, occupies the second part. 



Thirdly, it appears from the last investigation that one of the 

 possible forms of the curve is a straight line. It is easy to see 

 that it will be practicable to solve the equations for this form, 



