126 Mr. Challis 07i the General Equations 



cable to the motion of a particle which is affected simultaneously 

 by several independent causes, and it may be inferred that the 

 motion is the resultant of the several motions due to the 

 causes when they act separately. The equation applies, there- 

 fore, to any motion whatever. 



Suppose now the fluid to be of two dimensions. The equa- 

 tion for determining <p in this case is ^4 + T/i~ ^' 



Let ^^+/ = 7^. ^ = ^^-; ^=^'i + 



-' d X dr r dir dr r 



d ^ 



But llz = i^ + :f^ 



dr dr dt" 



rd(f 

 ~~d~ 



Hence — r-^ = 



rdr 



dr -J^'^l' J^- — ' 



f=At)\og.r + Fit). 

 The meaning of the result, vel. =-^, may be illustrated, 



as before, by conceiving the fluid to be contained in a cylinder 

 capable of expansion in the direction of the radii, and a small 

 cylinder of solid matter to be placed with its axis coincident 

 with that of the other. Let r = the radius of the solid cylin- 

 der, R of the fluid cylinder before the introduction of the 

 solid, S its extension afterwards, h = the common height of 

 the cylinders. 



Then tt r® /j = 2 tt R ^ S, ?• being very small ; 



And if the translation of each of the particles be supposed 

 to be effected by a uniform motion, 8 will represent the velocity 

 at any distance R from the axis. 



As the general integral of ^ + — = is also 



it is important to show that this is equivalent to the integral 

 above obtained. 



'^ = F(^+j/^-l)+/'(ar-j/V-l) 



di 



= A + B v'^:^! + A'-B'-Z-l; 



for 



