116 The Rev. Professor O'Brien on the Resistance of 



Now, R being very small, we have for a first approxima- 

 tion, 5 = — acosnt; and we may use this value of fi in the 

 second member of (2.), which gives 



n (a2 —a}^)^<2.ar R sin nt .dt; 

 <J 

 but, by the first of the experimental facts above stated, we have 

 0^ =su, where e is some ratio independent of a ; hence we find 



/'*n 

 R 

 - 



s\nnt.dt=:—n{\—&^),a,. . . . (3.) 



Now the resistance at any instant must depend upon the 



velocity of the ball at that instant, and most probably (or 



rather certainly) upon the velocities with which the ball was 



moving during a certain interval preceding that instant. In 



other words, the resistance may be supposed to be, in general, 



a function of the velocity and its differential coefficients with 



respect to the time*. We may therefore assume that R is a 



_ . . ^ „c?a dH dH . ... 



function 01 some sort of -7-, y-^, -j-g, &c., which we may put 



in the form 



R = P^ + P^. + P^,./ + &c., 



where P,„, P^/, P^//, &c. are homogeneous integral and ra- 



. , „ . pd^ dH cT^fl Q f , ,. . , „ 



tionai functions of -7-, -r-^, -r-g, &c. of the dimensions mfm',m.", 



&c. respectively, these indices being arranged in ascending 

 order. When the oscillations are sufficiently small, P^/, P^//, 

 &c. will therefore be inconsiderable compared with P^ ; in 

 fact, for very small oscillations, we may put R = P^ simply. 



Supposing then that the oscillations are sufficiently small, 

 we have R = P^ ; and therefore, putting for Q its first approxi- 

 mate value, we find 



R = T«'«, 

 where T is a function of t independent of a. Hence 



«/ 



Rs\nnt.dt='Sa,^i 







where N is some constant independent of a. 



Consequently it follows immediately from (3.) that 7n must 

 be unity; in other words, P^ must be a linear function of 



* To say that the value of the resistance at any time ^ is a function of 

 the velocity (v) at that time and the velocities during a certain interval 

 previous to that time, is the same thing as to say, that the resistance is a 

 function of j; and its differential coefficients with respect to t. 



