372 Mr. J. Prescott on the 



which, since / is greater than unity, is itself not greater than 

 4 

 — , and is probably very much less. There will be very 



little error therefore in omitting all powers of t beyond t' 6 . 

 Then 



Z 2 = (/-%(£j 2 {D + Cl0g e T + ^}. 



Now, by equation (95), 



_ 1 27tK 2 w __ a u 

 T ~f Ned ~u ~ f~u* 



where a is a constant independent of I and the same for all 

 similar shot of different sizes projected with the same N. 

 Therefore, 



- / -^W[D' + O'log.j + (J) , } > (106) 



where G is a constant that does not contain I, and has there- 

 fore the same value throughout the motion. The constants 

 C and D', however, change their values at the instant when 

 u = u 1} the change being such that Z 2 and its rate of increase 

 are continuous then. 



75. The constants for the earlier part of the motion, while 

 u is greater than u u are determined by the conditions that 

 Z 2 and its rate of increase are both zero when u = u . 

 These conditions clearly give 



D'=-l, C'=-3. 

 Consequently, when u >w b 



rhich 



/ 



Z 2 =^QpJ(^) 3 -31og|'-l} ; . (107) 



~ l Ql{leT-2>K-l} (108) 



by equation (5). Let the value of Z 2 when u — u x be denoted 

 by Z 2 ', then, from equation (107), 



Z 2 '=^^{@ 3 -31og^-l}. . (109) 



