358 MISCELLANEOUS STUDIES 



Equation (51) in which .H\ ( 1 -f- a 4 sin atf + O B cos a<) is substituted 

 for H becomes 



t 4 5 



Of OZ 



In the special case where 4 = a s = the solution is 



-f a 4 sin at + a 5 cos a*) T ~ =0 



(58) 



(2 2 \ / g 2 \ 

 t ) is an arbitrary function of it ^ ^ ) . This 



solution can be easily verified by substitution in equation (57). 



For a constant velocity H equals H v , the general solution of 

 the differential equation (47) is the sum of the three quantities 

 V + 0" + &" already found, and can be put in the form 



B(a l -\-a 2 nz}e-^-^ , , Ba s nz . B , Ba.nze- , , ) 



H t 



k(z-z n ) / 2 _ ~ \ 1 



~ffT7 ^ -- j~^J \ (59) 



Suppose the relation of the temperature to the time at a given 

 position 2 is known and that there is a constant horizontal velocity 

 H l from that point in any given direction. From equation (59) the 

 temperature at any time and at any point along the stream line 

 down stream from the point z equals z can be found by giving the 



arbitrary function / If ^- JL ) such values that when z equals z 



6 = tf -f- 0" + ff" will equal the observed temperature, which is a 

 known function of the time at that point. All of the constants in 

 the equation are given on page 347. Therefore f(t') being known. 

 when t', the time at the position, z equals z is known, the arbitrary 

 function is determined. For a time t and a value (z z ) of the 

 distance from z the expression 



f(t') (60) 



0-V) =(n (61) 



since, in general, the function is determined by the values of the 

 independent variable 



