1890.] Mr Sharpe, On Liquid Jets. 115 



will be taken so large as to be comparable with p. We will sup- 

 pose p so large that 



p . it p . Stt p . 5tt 

 - sin - , £— sin — , -f- sin — 



7T p 6lT p 07T p 



are very nearly unity. 



For some investigations connected with this subject it might 

 be necessary to expand these expressions in powers of 1/p, but 

 for our present purpose it will suffice if we suppose them actually 

 unity. To this end we must suppose p to be at least 22 (this 

 number making the largest rejected term ^th of what is retained), 

 but we may have to take p much larger than this. 



(3) and (13) then become 



a t + a s + a 5 = ,(21), 



a 1 + 3a 8 +5a 6 = (22). 



(16) and (17) become 



K + h + b s =pB (23), 



Q = 2A y + 2A 3 + 2A 5 (24). 



From (11), (21), (23), (24) we readily get 



A =pB = -Q=- 2A t - 2A 3 - 2A S (25). 



We will next suppose p so large that we may safely expand 

 the fractions in (18) and (19) in ascending powers of 1/p. To 

 this end p must be at least 50, and we shall get, retaining only 

 the most important terms in q H and r n , 



4C0S«7T , . _ . ar , . N ,„_. 



g, = - v (A 1 + 9A a +25A B ) (26). 



4cos?i7r, aH -,-,- x /«h,\ 



For large values of p, r n vanishes compared with q n . We 

 shall therefore retain only q n in equations (20). We shall further 

 suppose a v a 3 , a 5 to be small quantities multiples of 1/p 2 and that 

 their ratios are such that 



a 1 +3a 8 + 5a 6 = ..(28). 



(The object of these assumptions will be presently seen.) 



From (21) and (28) therefore we may put 



-a i = ia a = -a 5 =p 2 (29), 



where p 2 is a small quantity supposed to be a multiple of 1/p 2 . 



