332 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



C being an arbitrary constant, not containing t because t enters (p with a; only in the form 



Ct — «!. 



The last equation being again integrated gives 



X 



c - (p + \/{c - <py - Cn' = De^ ■ 



D being another arbitrary constant, a function of t such as makes the right-hand member 

 a function o( ct — v; 



D p Cn 



U 7T •-- " -7; 



But since ^ is a function of ct - x, this equation by introducing t, and properly assuming 

 the origin of t, may be written 



(et-j: ct-x \ 

 e~c" + e" c j (16); 



(Ct—X Ct-3f\ 

 e^ - e~ ^ ) ('7)- 



The last equation enables us to connect x and a ; for comparing it with (12) we have 



ct- X _ ct—x 



2 tan (wi( - a) = e c _e '-' (18). 



For a given particle a is constant, and consequently for that particle x so varies with t 

 as to preserve the truth of this equation. 



Eliminating x between (16') and (18), we get 



c — 11 = Cn sec (nt — a). 

 Now for a particle in the surface m = when t = t^\ 



.-. c = Cn sec {nt^ — a) 

 = Cn sec (ntf^ — nt,^) ; 



Cn h 



.'. — = cos {nt,^ — nt/^) = - from (14) ; 



C r€ 



. /- "A 

 . . Cn = — : 

 k 



1 ch , 

 consequently c - it = — sec (nt - a) (I9)j 



which gives the law of the horizontal velocity, as (11) gives the law of the vertical velocity of 

 a particle -. and it is worthy of remark that neither of these is represented by a sine or a 

 cosine. An assumption therefore that they might be so represented would be improper : and 

 from this assumption we may date in some degree the erroneousness of the results which have 

 been obtained by some writers who have adopted methods of analytical approximation. We 



have seen also that the argument ?it — a does not vary from to + - as some have supposed. 



Equation (19) shews that the horizontal unlike the vertical motion of a particle is wholly in 

 one direction, and is a maximum when the particle has reached its greatest vertical displacement ; 

 after which it decreases to zero. 



